I used the console command "showracemenu" to change the appearance of my character when my character was level 18, because I choosed a default face back then and now I didn't like it. Now several levels later at level 24 I realized cannot choose some perks I could before, because the implant (+1 skill) perks (that I still have and show at my perks list) are doing no effect at all now

Now for the questions:
- Is there any way to fix this? It really made me stop playing because I don't want to redo everything I discovered between those two moments.
- Is this affecting my other perks? (hard to check, unlike the implant ones, that can be deduced from SPECIAL)
- I read that using that console command may have blocked achievements for me. Achievements still seem to work for me... but will I still get the hardcoe achievement when completing the game?

I was playing perfectly until I realised this... now I don't even want to play until I find a solution, and reloading my lvl 18 save is not an option (too much things done in between)

The Showracemenus (and other ones that overhaul your character) can bork things as you have discovered. It is not guaranteed, but you must prepare for this. It has happened in every game since Morrowind which was on a different engine.

You do this by making a Forked Save and work off of the copy before you ever make any changes to your character in the console, even if you are just buffing up a SPECIAL stat. Absolutely before Showracemenu or Shownamemenu etc etc you need to fork a save. Then, play for a while and see what happens, good or bad. As long as you have a forked save, you can go back to pre-console condition and either try again or not.

I have a forked save, but like 8 levels ago I never realised this until now

Actually everything seems fine, except for that group of perks... so, there is a console command list to repair (maybe remove and relearn the perks) my agility, perception, luck and endurance implants?