Want to start blank project and transfer old into it. What problems can I expect?

Haven't tried that before, but IMO it's better to transfer shaders, then scripts, then art and then your prefabs. You can expect compiler error during the shift, but be sure that you don't miss anything, especially a shader or a script.

I don't think you will have to re-link your prefabs, only if you import them last.

If you're using Source Control, there are extra things to consider, like if you want to make a branch first, or what will happen if a file is moved vs. copied.

Project-specific settings such as collision matrix, custom tags/layers, build settings and so forth should be taken into consideration.

Well, I would go 'export' from the old project and 'import' in the new one - I always have bad experiences manually moving/copying stuff over I lose my prefab connections. I don't think you need to worry about any particular moving order this one (someone correct me if I'm wrong)