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Use datatemplate without listview

I would like to know if it's possible to use a data template and a list system without the listview controller on xamarin forms.

You can definitely roll your own controls that use the templating engine. It would just take a whole lot of additional work with custom renderers for each platform etc. If you want to look into that you can check out the source code for Xamarin Forms since it's open source to see how they implemented the ListView. I think the negative voting is because this is quite a broad question that doesn't have an easy answer.

How to apply Windows Forms Control to WPF Custom Control Library?

I'm trying to encapsulate Windows Forms Control (from previous version of project) to WPF Custom Control Library. Because I want to use it in future Wpf applications (like drag&drop from toolbox in VS). I successfully done this: WinForms in Wpf App. Do you know site/or tutorials where this things are explained?
What I got: .dll with WinForms control, Wpf Custom Control Library project (to place WinForms control) and Wpf Aplication.
Thanks for your help and suggestions.

What SonerGönül is refering to, is that if you read the links he points out.
You can see you should have done this steps better:
Do your homework
There is plenty of examples all over on how to achieve what you want to do.
Search on Google something like "how to host winforms in wpf"
You will get plenty of links like:
http://msdn.microsoft.com/en-us/library/ms751761.aspx
Even questions done in this site in the past.
Be specific, Make it relevant to others
You need to make a more clear question. Read your question as if someone else is the person who wrote it, and think if you would be able to understand it properly.
People in this site wants to see you have worked towards solving your problem, therefore you can mention things you already have done, and make a specific question.
Like "I implemented what this website suggest but I am getting this exception"
To answer your question
I really like this website: Mixing WPF and WinForms
Because it gives you a nice both side explanation of how Mixing WPF and Winforms works, with examples.

Any .NET Winforms control which look like this QT's QPushButton?

Is someone know any .NET Winform control which look like this button control?
QPushButton:
http://zetcode.com/gui/csharpqyoto/layoutmanagement/

You have several options here. You aren't specific about exactly which part of the platform you are using, but if using Winforms, you can certainly customize the buttons to some extent.
If you are using WPF, you can pretty much make it look exactly like you want in XAML. Check out Expression Blend.
As #Dimitri put it, the sky is the limit, but you may need to do the leg work.

You can create custom controls according your need and have its reference added to your project. you will have it added to your toolbox that you can use.

If you are refering to a button that is located on this example form:
We currently finishing our own application that has rather similar looking buttons. We did this by using a third party component. Steps are:
We purchased DevExpress.com's WinForms library.
We developed our own DevExpress Skin (with the help of an external screen design guy)
This was a bit of a work and some amount of money but the results look pretty neat.

Its possible to modify a datagridview to this?

I need to create an interface similar to this mock-up (sorry, drawing is not my thing). It was impossible for me to create " the component" in the second column, the one with the link-buttons and and the formatted text. Is there any way to do this or some third-party component to do something similar?
The mock-up

This should be easily possible using Template Field (this is assuming that you are talking about ASP.NET application)

I think that you're looking into the DataRepeater control
It is part of the VIsual Basic Power ToolPack but it is a .net control so it will work from C# as well.
Here is a detailed video showing an example of almost exactly what you're trying to accomplish.
http://windowsclient.net/learn/video.aspx?v=30534

Drag and Drop control for Silverlight

Recently I needed to add drag & drop functionality to a Silverlight application. Can anyone recommend a good drag & drop control?

I created a Drag/Drop controller that I think works really well. I have been using this technique for a while, and I have been very happy with it.
http://houseofbilz.com/archive/2009/02/10/drag-and-drop-with-silverlight.aspx

Here is a link to the best one I have found so far: http://nickssoftwareblog.com/2008/10/07/silverlight-20-in-examples-part-drag-and-drop-inside-out/
The code is available as a download from the blog post, although you have to rename it to a .zip: http://nickssoftwareblog.files.wordpress.com/2008/10/genericdragdropzip.doc

You can try Blacklight Controls which have a really neat drag-dock panel control. Here's a link to a blog describing how it works.

I recommend the Silverlight Toolkit for dragging and dropping elements within the same application.
It already has been updated for Silverlight 5.

I am guessing you have already found a solution by now. But here is a solution I wrote and am currently using which is very flexible and easy to use: http://sl4dragdrop.codeplex.com/
It works on SL5 very well, all you have to do is add dependency properties on the items or itemscontrol on which you want to enable dragging or dropping.
By default, that is enough. But if you want to customize parts of drag/drop, it gives you interfaces to implement that can control things like Element to drag, Which element to drag from, What image to show while dragging etc.
I have recently refined it a bit and will be posting an update soon after some more testing.