I have heard Norwegian people say "fjesen" for face - is this word acceptable?
This is weird. So if you use vasker as a reflexive you need 'i'? I typed 'Han vasker seg ansiktet med såpe' and it was wrong. Do you really need to say 'he washes himself in the face'?
Yes, you do need the 'i' when you use the reflexive version of the verb. It gives direction to the act, and omitting it sounds just as strange as it would to omit 'in' in English:
"He washes himself the face with soap."
If you want a shorter version you can use the non-reflexive version of the verb:
"Han vasker ansiktet (sitt) med såpe."
but can it mean that he is washing someone else face? for example he is a barber.
Staying as close to this sentence structure as possible, you could substitute the object form, as the action is no longer reflexive, but targeted at that other person: "Han vasker ham i ansiktet".
However, it would be more common to rephrase the sentence with a possessive when washing someone else's face: "Han vasker ansiktet hans."