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Which framework best to use for a cross-platform cross-devices platform with multiple apps

I am working out an idea for some applications.
The applications needs to run on:
Windows as a desktop / UWP app
Android as a mobile app
IOS as a mobile app
Windows Phone as a mobile app
Website as a web application
All applications will have the same functions.
All applications will need to view, edit and store information in and from a database.
I used to program in VB back in the days that we only had windows forms and I am new to the nowadays frameworks.
Till now I was thinking about creating a webservice to let all applications communicate with the database and then create all the other applications in their own framework / languages.
I have found some articles about ASP.NET WebAPI which looked like what I needed for the webservice but I don't know if I can also use that for Android and IOS.
I also found something about Xamarin for building apps for all the platforms I am targeting.
Could anyone maybe inform me what framework best to use in this case.
Should I use Xamarin for the client apps? or is it better to create the apps in their own framework per platform?
Should I make the webservice with ASP.NET WebAPI? or is there an other framework which will be better in this case?
I also found something about Node.JS but am not sure what its for.
I am asking this because I need to educate myself in the languages / frameworks before I can start programming and I don't want to start educating in everything and in the end find out that it is useless in this scenario.

I´m using Kaa Platform(https://www.kaaproject.org/) to manage all comunication between devices ( ios, android, WP, etc ).
And for developing, i strongly recomend:
if you will use just android and ios: react-native ( https://facebook.github.io/react-native/)
if you will use android, ios and others: ionic ( https://ionicframework.com/)

Running OpenCart in my mobile application

I'd like to know if it's a way to write a mobile application which can purchase products through OpenCart website. Here are two possible methods I can think of. Please suggest which one is better or if there is another better way:
OMF - OpenCart mobile framework. I guest it's just a theme for the open cart website. Is it possible to run this inside PhoneGap ?
OpenCart Restful API. We can build the native UI in mobile application. But if there are any limitations?
I'll appreciate if anybody who have experience to build the open cart website for the mobile application advise. Thank you so much.

I shall answer because I was working on somewhat similar project before. Your suggested solutions first.
1] Using phonegap you develop hybrid mobile apps which run on webview on respected mobile platforms. And I don't recommend this method because what it does is renders the exact website DOM in mobile's webview and it will get rejected by Apple since they won't allow this due to native design guidelines consistency. So, unless you are only targeting Android users, forget this option.
2] Building native mobile app is preferable. However I should give you a heads-up that Opencart has prewritten APIs for Orders, customers, shipping, vouchers etc but not for products. You will have to manually write it.
I can't figure out what limitations you are referring. Obviously, in a mobile app the desktop version extensions won't work. the requirement mentioned in the first line of question can certainly be meet using API with native development.

Phone Gap or native Android wrapper?

I want to develop an application which works both in a regular browser and as an Android app. Both seem suitable but would I be right in saying if I went with Phone Gap, deploying the same html content as a regular, server hosted website is not trivial and would require modification? Also, are multi page applications not desirable in a Phone Gap wrapper?
Are there any alternatives I should consider? I'm surprised someone hasn't already written an Android wrapper with all the hardware access wiring done already. Or have they?

There are few things you should know,
If you are going to publish your app also on iOS, you can't use server hosted website for your app.
All of your basic functionality must stand on your app assets, is one of the iOS guidelines, and its also make-sense for best ui performance (both iOS & Android).
Use server-side wisely, when you want to fetch data which have to be up-to-date.
there are some new mobile browser features which will be available in the future, that allow your website to use the device hardware functions:
Device API on W3C
Good luck dude ;)

You can try with jquery mobile framework:
http://www.jquerymobile.com
It is jquery based framework for developing web apps that works on standard web and on multiple mobile devices (with cordova)

Web app targeting iOS, Android and WP7?

I'm about to start on a new web application project where I need to support iOS, Android and Windows mobile 7. Is there a framework that will support all three?
In the past I've used Sencha Touch for iOS and Android, but it would be nice if there was a framework for all three systems.
I would like it to be as native looking as possible since this is a demand from the client.

I will use PhoneGap for this. You can refer official PhoneGap site at http://www.phonegap.com/
It supports six platforms which includes ios, android, blackberry, windows, symbian, palm os..
You can refer complete chart here for different features in different mobile os http://www.phonegap.com/features
You can see for Windows its still coming soon.
https://build.phonegap.com/
However they are updating with Bada & MeeGo..

W3C has a list of mobile web app best practices. Please check this link
http://www.w3.org/TR/mwabp/
I hope this will help you in designing a mobile web applicaiton.

Although still in alpha you may want to consider using jQuery Mobile http://jquerymobile.com/ for the UI. It does have some WP7 support, but expect this to get a lot better after the IE9 mobile/Mango update.

Best platform for programming to IPad and new Windows and Android Pads

would like to build an app that can run on any of the new PAD's hitting market.
I don't want to limit our users to a specific OS.
What is the best solution to allow all these pads to use our app.
The app needs to be able to run offline.
Thanks!

You could write a web application and use HTML5 for off-line caching.
http://diveintohtml5.ep.io/offline.html
Alternatively, you could try and use Titanium Mobile.
http://www.appcelerator.com/products/titanium-mobile-application-development/

You should also spend some time reading this discussions:
is-there-a-multiplatform-framework-for-developing-iphone-android-applications
technology-to-write-iphone-blackberry-and-android-phone-at-the-same-time

Have a look a MonoTouch, it will let you write all your logic in C# over all the main mobile platforms. However you still need custom UI code for each platform, but as least it will all be in C#.
(Better then having to use C#, Java and Objective C)

You could use Adobe Air that could works in most of Mobile/Tablet operating system Iphone,Android and Windows 7. but im not sure if Apple will approve your App if you plan to released to the App market.
Another way to think about it is to create an HTML resources and then integrated in away seems native to the system its more work but you will have a higher chance to get approved from apple and the app look more integrated with the OS .

The obvious -- and currently free -- answer is to use Adobe FlashBuilder to develop iOS apps. There is an iPhone and iPad emulator included. This does not use xcode, but you get most of the features to work with, and you can also develop Android apps from the same set of code. Further, with minor modifications for mouse usage, you can also have the apps run on any desktop as an Air app.
Adobe's website has detailed directions for how to create iOS apps on Windows with Adobe Air, though the most useful instructions for Air are from untoldentertainment.com.