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When to use Web Api in MVC

(First i just want to say i'm sorry if its not worded the best but i have been researching for hours and i thought maybe someone on here could clear this up for me)
Im new to creating web apis and i have been googling just doing some research and i have built a few MVC applications just to get exposed but i never really thought of making an API for it until today. One of the reasons its listed to be useful is it allows your application to be used across tablets and smart phones etc. What im not understanding is how you would do this as in would you just add something to the API so it makes it compatible for all browsers or if i need to re build it using Web API instead of MVC.
Thanks,

I think your looking at a Web API from the wrong perspective. It's not really about compatibility necessarily, but rather about the ability to reuse the code/back-end functionality.
So rather than having your dependencies all wrapped up in one MVC project, the references are external. This allows for essentially the same functionality across multiple projects as long as the request are being handled in the same manner.

How can I run ASP.NET MVC within a Windows Service

I'm working on an application that requires me to host a WCF REST Web App using Windows Service. Now since it is going to be a bit more complicated than just a small API, I would like leverage MVC capabilities as a lot of things would be lot easier if done using the 'Controller' way (if I can say so). I'm don't have deep knowledge of MVC and Windows Service.
So far, I've found only two links here & here, that are somewhat related to what I want, but still not there. Could anyone please point me to a working example or create a small demo? TIA.

Based on your comments, it is clear that you are wanting to run REST-based web-services, self-hosted, on both Linux and Windows.
The recommended way to do this is to use the new ASP.NET Core platform, running on .NET Core.
Microsoft provide a good tutorial here:
https://docs.asp.net/en/latest/tutorials/first-web-api.html

Another link just received from a quick google search Here but yes need some more clarity on how you plan to run it or what its for.
We used Nugent package manager years ago for self-hosting a web API

Which API to use for Dropbox files manipulation

I am developing a winform application.I want to provide support for Dropbox in my application. Since my application is .net 3.5 based. I can't use the Dropbox SDK. So, I will be using the rest API. But there are two sets of Dropbox API V1 and V2 for Dropbox. So, are V2 API's stable enough, so that I develop my application using them or should I use the V1 API's.

Answering your question is not that easy since you have to factor in a number of considerations you do not provide information on. Skill level, product time constraints etc all play a role here.
That being said, I have used API V2 but there are some teething issues at times and support / updates are not always lighting quick. API V1 on the other hand have plenty of examples and community support is readily accessible. So my answer would be to use V1 if you favor stability and support over being on the latest release. Even if you use V1 I dont believe you have to fear the API being depreciated very soon, Dropbox has in the past given plenty of notice before they drop support for a API version.

google Maps Api vs Bing Maps for C#

I want to use one of this services for maps in my website. Which do you think I should go for provided that the application is writen in C#. Which is the best SDK for working with this services in C#.

use great maps api instead it is an open source project and very useful. You can change the maps from bing,yahoo to google maps.Of course you should take the api key for google and bing. The documentation is not enough but i think it is very easy to use. Hope this helps.

This is somewhat subjective; however, go for what you believe would provide the best service to your "customers" (whoever they may be). I would not let your determination be based on what "SDK is best" per se, as you can find a hundred examples out there to do both bing maps and google maps. As far as popularity, I would imagine google maps takes the cake:
Try this google maps example: here

I suggest taking a look at this:
http://www.jonasson.org/maps/
I had to make the same decision a couple of years ago, and when using this it was clear that Google Maps had much better satellite imagery, and road data than Bing Maps (specifically in the locations our customers would be using it)
The Bing Maps API is coded in a very familiar .Net fashion. Although its all JavaScript, the Google API feels more like a Java implementation (as you might expect). This take some time to get your head around, but shouldn't really be an issue.
I found the Maps Group on Google excellent as a resource for research and queries.
UPDATE:
Google released v 3 of their API recently and its supports street view as part of the maps API. Pretty compelling reason to use it.

For my needs (and yours may well be very different) the parts of the API I use are easiest to access via Javascript in both cases. The big win for me in the app I'm working on right now is the Silverlight control for Bing maps, and the ease of integration into a WP7 app.

How can I use OpenStreetMap in my Compact Framework App?

Does anyone know, how I can use OpenStreetMap inside my Compact Framework application? Is there a Framework or something like that? All I can find in their wiki is how to contribute to their project and to user their software to map data. But I want to use their maps to show the users location inside my own app. I could not find anything about using their web service or whatever I have to use to show their maps inside my application.

There is a project on Google Code that wants to create a .NET library for the OpenStreetMap API. The website states that there currently is an alpha release, so it might be worth checking that out: code.google.com/p/openstreetmapnet
And there are two similar projects on CodePlex as well:
osmscout.codeplex.com
gmap4dotnet.codeplex.com
Maybe those can be used in the Compact Framework as well, or you could ask the project teams to add that feature (or get involved yourself, it is opensource after all).

CloudMade provides APIs for embedding OpenStreetMap maps in many languages - but not .NET as of the time of writing. There are plain HTTP APIs for map tiles available that you could call from .NET programs, but this is of course fairly low-level.
The CloudMade APIs are designed for developers who want to embed OSM output in their projects, rather than the OSM API which is geared towards map creation, so I think you might be interested in the CloudMade stuff.
See http://developers.cloudmade.com/projects for more.

They don't appear to have any CF libraries, but the API specification is well documented, so there's no reason you couldn't integrate with it.

I've developed a CF component that can download OpenStreetMap tiles, and also plot points of interest and basic routes. Depending on your requirements, and any restrictions on licences that you're permitted to use, then it may be of use to you?

Brutile looks quite promising. It is a library designed for C# to connect to OpenStreetMap sources. (Silverlight demo available Here.
The wiki also has a listing of all .NET related pages. Of the documented projects, OpenStreetMapViewer looks like it fits your needs the best. It's designed to display a portion of OpenStreetMap within a .NET app. I'm sure at least one of these will work for windows mobile.