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Which is better to handle SQLite between Exposed and Anko in Kotlin?

I know that both Exposed and Anko can operate SQLite easily, could you tell me which one is more better when I develope an Android App?

Use the framework you are most comfortable with. I tried Exposed once and it was ok for what I've used it. jooq might also be a valid alternative instead.
Reading the documentation of both, Room and Anko, I would stick to either one of those.
Regarding which of those, I just found an issue asking the very same question (even though just as a second or third question):
https://github.com/Kotlin/anko/issues/484
Some other resources I found regarding Room and Anko:
Stress-free SQLite with Anko: the comment regarding Anko vs Room is also insightful:
Anko SQLite provides a nice API to manage your data persistence layer but you still have do the heavy lifting by yourself. While Room is more like a framework. Generates databases from annotated classes for you, provides observable queries and has a really nice testing support. Also works well with Android Architecture Components.
Ah... and don't get me wrong. Exposed is ok too, I am still using it ;-) But if you are familiar with both, you may also have your preferred choice already. If you don't know any, you can try both and choose the one where you grasp the documentation more quickly or you feel more comfortable sooner.

What is Fastest way to query a database android?

I have a database that contains 35000 records, and can easily get up to about 60000 records,
What is the best way to query these data? Will ORMLite or GreenDAO be usefull? Or should i just stick with Cursor?

That depends on what you mean by "best".
Fastest, most flexible, best integration with Android library functions: Stick with Cursor. Since ORMs are just wrappers around the native Android cursors, they are unlikely to be faster than direct access. In many cases, the Android UI library classes are optimized for Cursors (e.g. SimpleCursorAdapter, which allows you to bind a cursor to a UI list).
Most readable, most maintainable: An ORM might help here, but that depends on a lot of factors, including your project complexity, your application architecture, etc.
See the following programmers.se question for more information:
Does it make sense to use ORM in Android development?

Does Android support any other database system other than SQLite?

I have been searching and have found out, that android supports only SQLite databse and no other. Is this true?

Yes. The Android library provides native support to only SQLite. Of course, this doesn't mean you absolutely can't use other databases on Android; if you need to use other databases, you'll have to either look for already-existing third-party libraries(1), or roll out your own API.
(1) Careful there: If you look for third-party libraries, make sure they're built specifically for Android, since Android includes only a subset of the Java standard library. If they're not specifically built for Android, there's a possibility that the libraries won't work due to missing classes.

Yes, Android Supports H2 Database too. please check it out with below link,
http://www.h2database.com/html/tutorial.html#android
Thanks,

Though SQLite is natively supported on Android and is most used database, there exists other options as well. Listing a few of them below;
Realm:
Reactive, concurrent, and lightweight, allowing you to work with live, native objects.
https://realm.io/docs/
H2:
Full Unicode support including UPPER() and LOWER().
Streaming API for BLOB and CLOB data.
Fulltext search.
Multiple connections.
http://www.h2database.com/html/tutorial.html#android
CouchDB:
Full CRUD and query functionality, NoSQL, lightweight, embedded, syncable
https://developer.couchbase.com/mobile/
LevelDB:
Lightweight and single purpose (not an SQL database)
http://leveldb.org/
Java wrapper: https://github.com/hf/leveldb-android
I am not including BerkeleyDB here as (AFAIK) it needs OS level changes to be made to replace the SQLite routines. More info here https://blogs.oracle.com/berkeleydb/now-you-can-build-berkeley-db-into-your-android-apps

SQLite is the only one I ever actually see being used. I would suggest using it because it is lightweight and free to use. When developing your App, lightweight is key, memory is limited on these devices! Is there any specific reason why you wouldn't want to use SQLite? Any feature that you're looking for that you don't see in SQLite?

Android application and myBatis

I'd like to use myBatis (iBatis 3) in an Android application. Has anyone tried such a thing or know of any resources for this?

aBatis is a data mapper framework available for Android
　that couples objects with stored procedures or
　SQL statements using an XML descriptor or annotations.
aBatis is like an Android equivalent of iBatis.
---simple & light ORM library like iBatis for Web development
--carrying ibatis's feature
--easy-to-use as iBatis
--shorten a development period
--independent of development phase
--Android sdk1.6 and up
http://sonixlabs.com/abatis/

The first thing to do in order to do that is to compile myBatis for Dalvik. But it's likely to be too heavy for a device like a smartphone.
If you are looking for a lightweight persistence layer, you could look at Ammentos:
http://www.ammentos.org/
Same trouble: you will need to compile it for Dalvik.
Existing ORMs for Android:
http://ormlite.sourceforge.net/sqlite_java_android_orm.html
https://www.activeandroid.com/

It's better to use lightweight lib in android apps like greenDAO or Ormlite, greenDAO is an open source project to help Android developers working with data stored in SQLite. SQLite is an awesome embedded relational database. However, developing for it requires alot of additional work. Writing SQL and parsing query results are quite tedious tasks. greenDAO will do the work for you: it maps Java objects to database tables (often called ORM). This way you can store, update, delete, and query for Java objects using a simple object oriented API. Save time and focus on real problems!
greenDAO’s primary design goals
Maximum performance (probably the fastest ORM for Android)
Easy to use APIs Highly optimized for Android Minimal memory
consumption
Small library size, focus on the essentials

I create simple android project that using mybatis
check this: https://github.com/gustaroska/HijrDroid

Higher level database layer for Android?

Are there any good database abstraction layers/object relational mappers/ActiveRecord implementations/whatever they are called for Android? I'm aware that db4o is officially supported, but it has quite a large footprint and I'd rather use a more conventional database (SQLite).

I am the main author of ORMLite which was designed to be small[ish] but still provide higher level functionality. ORMLite makes calls to the native Android OS database APIs to support its ORM functionality. See the following for general information
http://ormlite.com/sqlite_java_android_orm.shtml
Here are some Android example applications:
http://ormlite.com/docs/android-examples

I tried the Sugar ORM, which is very basic (and easy to use) but it worked for my needs.
Sugar website

There is an 'android-active-record' project which provides ActiveRecord abstraction for accessing Android SQLite database.
It's available here: http://code.google.com/p/android-active-record
It allows to eliminate most of boilerplate coding when performing CRUD operations on database entities and also minimizes efforts for creating/maintaining a database structure

Try ActiveAndroid. It is free and open source (Apache Version 2.0).
From the website:
ActiveAndroid is an active record style ORM (object relational
mapper). [...] ActiveAndroid allows you
to save and retrieve SQLite database records without ever writing a
single SQL statement. Each database record is wrapped neatly into a
class with methods like save() and delete().
[...] Accessing the database is a hassle, to say the least, in Android.
ActiveAndroid takes care of all the setup and messy stuff, and all
with just a few simple steps of configuration.

If performance and size matter, you should have a look at our open source ORM tool greenDAO. We wrote it because we did not want to compromise on speed. Other tools heavily rely on reflection, which is very slow on Android. Despite the tiny size (<100k), it supports relations, query builders, etc.

Shameless plug, but I've been working on a new open source Android framework called Infinitum. One of its main features is an ORM which has a criteria API similar to Hibernate and a few other nifty features (associations, lazy loading, etc.). It's still in its early stages, but I think it's coming along pretty nicely.

I have written a new ORM, for android, that's aimed and being as easy as possible to implement. It support lists and SQL free migration a couple things which I always found had an overhead in other libraries.
http://www.rushorm.com/

I faced the same problem and looked at both android-active-record and ActiveAndroid. I found android-active-record didn't handle the things I cared about (relationships for example), and ActiveAndroid isn't free. Therefore, I decided to write my own library. It's called AndroidRecord and it's hosted on GitHub and you're free to do with it what you want (I think I'm going to go with the MIT license). I use this every day and I'm content with it, but I'd love to get feedback.
If you need to know how to use it, I'm working on the documentation. If you need it right away, you can check out this lame example project which should be enough to dip your toes in. You can also email me of course.

There's also Neodatis and Perst (Lite).
I've toyed with Perst a year ago and concluded it's not worth it.
After all, a) Android runs on a rather restricted device with ~16mb of heap space per app and b) You customers would really appreciate performance and low power consumption.
So my advice is to go with SQLite and hand-written SQL. It's not hard at all and the wrappers provided by Android SDK are really nice.
EDIT: In 2012 the advice would be to use the ORM component of DroidParts (which is my project).

I was comparing basics of ormlite and greendao some time ago. You might want to take a look there. I plan to write some follow up with more advanced stuff in the near future but for now it's only a basic stuff. In my own project I'm using GreenDAO.

Have a look at Androrm. It is open source and well documented (see here). If you ever worked with django, you will notice, that the syntax is very similar.
Androrm also supports abstraction classes for the most common field types, plus relational fields. This way it enables you to query for your data in an very easy manner with only very little effort on your side.

SQLite is explicitly part of Android:
http://developer.android.com/reference/android/database/sqlite/SQLiteDatabase.html
However you might have to create your own abstraction layer (query builder for simple queries), or otherwise deal with SQL.
Maybe http://developer.android.com/reference/android/database/sqlite/SQLiteQueryBuilder.html is what you need?