"Can you rewrite "mă spăl pe față" as "îmi spăl fața"?"
Yes, you can - mostly depends on the context and any nuances you want to bring to your sentence. The most likely way to make/write that statement is "mă spăl pe față" ( I wash my face). BTW, I'd use the same English translation for "îmi spăl fața", without additional context.
"Also why does the verb a-și imagina require și? Why not just say a imagina?"
In that context,
a imagina is used as a reflexive verb because the action is directed towards the subject which performs it. If you were to say
eu imaginez/închipui c-ai plecat it won't sound natural (eu îmi imaginez/închipui c-ai plecat).
On the other hand, the verb
a trece can't be used that way, it's wrong to say
(eu) îmi trec strada (I cross the street) as opposed to
eu trec strada*. When crossing the street the action is directed towards the street, when I imagine (things) I imagine for/to myself.
(*) I cross my street is
eu trec strada mea. Not to make things too complicated but this correct:
Îmi trece prin minte (here is a thought//I'm thinking of this... )
For your reference here is the dictionary link to the verb
a imagina from
dexonline.ro.
There is a lot more info about pronominal and reflexive verbs
here - I'm assuming you'll be able to follow the text in Romanian.
Note: CD/CI - Complement Direct/Indirect
Here is a list of several reflexive verbs in Romanian from the link (almost all can function, with a somewhat different meaning though, as standard verbs by removing the reflexive pronoun):
"
Verbe reflexive: a se ruga, a-și reveni, a se bucura, a se mira, a se gândi, a-și închipui (my note, a-şi imagina), a se strădui, a-și bate joc, a-și da seama, a-și aminti, a se odihni, a se dezgheța, a se apuca, a se ambiționa, a se plimba, a se trezi, a se teme, a-și uita, a se veșteji, a se numi, etc."
