To answer the first question, yes, James "Logan" Howlett can get tattoos, but tattoos are pretty much ink getting scarred into your skin. As such, the ink will get ejected and the scarring will heal for Wolverine. He explained this to Storm in Ultimate X-Men– which is admittedly not regular Marvel-616 continuity.

Can Wolverine get drunk?

In Wolverine #900, Logan revealed his healing factor processes alcohol nearly as quickly as he can drink it, with a bottle of whisky only getting him tipsy for seconds. It seems Wolverine is capable of getting drunk, but only for an extremely short amount of time (unless he has a constant supply of alcohol.)

Does Logan have tattoos?

Logan Paul only has one tattoo, the design for the tattoo that he chose is a character from the Japanese franchise Pokémon, named Squirtle. He documented himself getting the tattoo in a vlog.