Gaming is for having fun. Fun requires clear rules.
1.in explore action
"the hex can be placed only to the spot that you explored"
Is that mean I have to claim the spot that I want to explore and can't change it after I turn over the hex?
Yes, especially that you have to use right pile.
2.in infulence action
move your infulence disc "to a hex that does not contain an influence disc or an emeny ship"
Can I move my influence this to a hex that contains a ship of my diplomatic relations country?
If I can do so, is that a breaking relations from him?(He put a ship in the hex has my influence disc)
He's still your enemy, so no.
3.the "Artifact Key"
"You must immediately take 5 Resources for each Artifact on your hexes."
Also what is "your hexes", the hexes have my infulence discs or the ones have my ships is also my hexes.
"in groups of five" means if I got 5 Artifacts, I can add 15 Money, 5 Science and 5 Materials. Right?
Only your discs. Any combination of all types, just keep the sum equal to number of artifacts * 5.