I would like to achieve something very similar to Microsoft Access query designer – I am talking about a plane(canvas?)-like surface on which users can place and move controls. Is this even possible?
If it isn't possible with free .NET controls – then are there any paid ones, which offer similar functionality?
Anything is possible if you try hard enough.
I would explore WPF, it will provide capabilities to custom build your own controls and will get rid of that tedious work of keeping the UI updated.
Make sure to follow a good UI pattern if you do end up working with WPF, here is a good video that explains how to implement MVVM in WPF.