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close fragment and activity B at the same time (go back to activity A)

I am working on an application with specific architecture regarding navigation. The situation I have right now is like this:
I have activity A. My activity opens activity B with function startActivityForResult().
After that on activity B is added fragment X.
Right now I am on fragment X and I detected the close button click and I transferred this event to activity B. Now I want to go back to activity A. The only way I see at this moment to achieve this is by calling 2 times function onBackPressed() in a row. This creates laggy animation in which the user sees removing of fragment and activity B in delay. This is bad for UX and looks bad in code.
Does anyone have some better suggestions?

on your fragment X button click.
button.setOnClickListener{ (activity as ActivityB).finish() }
or
button.setOnClickListener{ requireActivity.finish() }

Preload Activity in backstack

When user click on a notification, I set up a backstack of Activities A -> B, where B is on top and shown to user. I would like the lifecycles of Activity A to run, so that when user presses back button and comes to Activity A, it is already ready. What could I do to achieve this?

This cannot be possible as android will not allow us to do that, the only way I can see you can achieve this by doing your Activity A operation inside your Activity B And provide those details to Activity A when the user pressed back button.
Note: If you are showing any kind of list on Activity A or doing similar kind of work, you can have one Singleton Class where you can get your data in Activity B which required by Activity A, and provide same data to Activity A when the user pressed back button.

You should open the activity as normal but then put this line of code in the OnCreate() method to bring to the back.
moveTaskToBack(true);

Activity launchmode and lifecycle

I have two activities A & B. in A, i am showing a list of titles, and on clicking a title, it will open the detailed article in activity B.
I declare A as singleinstance in Manifest.
But if I declare A as single instance, and when the Activity B is opened and paused, then Activity A is not available on backstack.
I will try to explain by reproducing:
Activity A (launchMode = SingleInstance) with list of titles.
On clicking a title, Activity B opens
On clicking back button/up navigation, Acitiviy B finishes and Activity A resumes.
Again open activity B.
Press home button of device (Activity B goes to background - onPause)
Activity B is available in Recent Apps
Open app from recent apps/launcer - Activity B opens
Clicking back button/up navigation - Act B finishes, but Act A not resumed.
How can I provide better up navigation?

For my idea, you can change lauchmode of Activity A and Activity B from Singleinstance to android:launchMode="singleTop". I work fine for me. It is well said here. Let try.

When you will press the home button the activity B would call onStop() method. For a better understanding you can refer to https://developer.android.com/guide/components/tasks-and-back-stack.html

Android how to save and restore state of Activity when calling finish()?

In Android, from Activity A, if I start new Activity B, Android will automatically save all the state of Activity A. So when I click back or call finish() from Activity B, Activity A will be restored to the state when I start Activity B (for example: position of a ScrollView in Activity A, or the value of ProgressBar). In Activity A I have a ProgressBar showing progress of some task, then if I come back from Activity B, the ProgressBar is restored and keep running.
Now what I want is, from Activity A, I will call finish(). How can I save all the state of Activity A just like Android did (include the ProgressBar as I describe above), so when I start Activity A again, I can restore everything like before calling finish(), and my ProgressBar keep running?
Can I push Activity A to something like stack in Android and then pop it out? Anyone know how to achieve this?

Do not call finish on clicking the back button.
Instead call the method "moveTaskToBack(false)". This will 'minimize' your application.
Next time you open the application the application opens with the previous state.

The whole point of finish() is to say that your Activity is "done and should be closed". If you aren't done, just don't call finish().

When you go from Activity A to Activity B, Android does not kill Activity A. Activity A still lives in the background. It can however be killed at any moment if Android needs the memory.
To save the state of an activity you have to do it manually.
Here's a similar question with a good answer: Saving Android Activity state using Save Instance State

If you want to finish your activity and then start it again just to load some new configurations, You can call activity's recreate() method instead of calling finish(). Calling recreate() will call onSaveInstanceState() and onRestoreInstanceState(), helping you to restore your previous configurations..
You can look at this answer for more detail : How do I restart an Android Activity

How to save instance of an activity?

I am new going for the android.I am working on an app in which I have two activities let say A and B.
In activity A I have a list view of some items.I have a button in activity A which takes me to activity B.In activity B I have a seek-bar.
I am using the seek-bar to filter the result of activity A.
I have two buttons in activity B cancel and filter.
After adjusting seek-bar if user clicking filter button than it takes user to activity A and showing filter results.
User can play between activity A and activity B.
Now three different scenario are there for coming back from activity B to A.
By pressing filter button
By pressing cancel button
By pressing phone's back button
After adjusting seek-bar if user pressing filter button then activity A is re ordering and showing filter results. Here I want to save the instance of activity B. so that from activity A if user again going in activity B then I can show the previous state of activity B.(I am able to do this)
In second scenario if user adjusting the seek-bar again and pressing cancel button then Activity A is reordering.Here I do not want to save the instance of activity B and if user again going in activity B from activity A then I want to show the previous state of activity B.(I am not getting how to do this ??)
In Third scenario if user adjusting the seek-bar again and pressing phone's back button then Activity A is reordering but now if user again going in activity B from activity A then activity B is restarting that I do not want, here also I want to show the previous state of activity B.(I am not getting how to do this also ??)
I am stuck with this problem.
Thank you so much in advance.

You must consider each of the cases and manage the activity lifecycle accordingly. You did this just fine, the problem is how to manage it. So, the first step is to study this:
activity and lifecycles
After understanding how the lifecycle of activities is handled by the android os you're on your way: Manage the life of your second activity so that state is mantained by overriding the onPause method (which is called when your activity is no longer in the front of the application) or finish the activity if you don't want to save the state (effectively calling the onStop method.
I would solve this in this way:
in activity A i call the activity B with a startActivityForResult. This way, when the activity B is finished, it's state is not mantained. So, call finish() inside activity B to return to activity A without saving it's state.
When wanting to save the state of activity b, call the activity A so that the onPause method gets called and the state saved.
Hope this clarifies it for you.