Does it matter in which country I obtain my EASA Part 66 AME licence?
When you have to deal with the authority, the theory and the practice varies a lot. So, there are things that theoretically you should be easily able to do, but that the authority won't accept, for no real reason. Sometimes it's just how the guy at the desk woke up in the morning. 

Anyway:

1) Yes, you can apply in any country of the EU28. I know Switzerland won't accept your appliances unless you work in Switzerland, don't know about Norway and Iceland. Have no idea regarding the new situation in UK after the Brexit. 

2) Because Europe is not a federation of States like in USA, and each authority want to impose own power and rules and protect our market. Ask them why.

3) Theoretically no. Pratically, maybe. 

4) All EASA countries. Where do you work? Where did you get your theoretical modules and practical experience? Try starting from there.

5) Repeat the question again, it's not very clear to me. 

To apply you need to compile and send the EASA Form 19 (You can find it here) to any European competent authority. 

You can also find the Annex III, the Part 66 of the Easa Regulation 2042/2003 here, with all the pertinent information. (In particular check the 66.A.10 Application)