Learn Swedish: Han tvättar sig.

sig is a reflexive pronoun used when a third-person subject (like he, she, it, or they) does an action to themselves. Because the subject here is Han (he), sig translates to "himself".

Could I say Han tvättar honom instead?

No, that would change the meaning entirely! If you say Han tvättar honom, it means "He washes him" (as in, he is washing a completely different guy). You must use sig to show that the doer and the receiver of the action are the exact same person.

Do I always use sig for "myself" or "yourself" too?

No, sig is only used for the third person (he, she, it, they). If you wanted to say "I wash myself", you would match the pronoun to the subject and use mig (Jag tvättar mig). For "you wash yourself", you would use dig (Du tvättar dig).

Does Swedish have a way to say "He is washing himself" right now?

You are already looking at it! Swedish does not have a separate "-ing" tense like English. Han tvättar sig can mean both "He washes himself" as a daily routine, and "He is washing himself" right at this very moment.