Mi puo fare o Puoi fare per me

Do these sentences both translate as "Can you take photo for me?":

Mi puo fare una foto?

Puoi fare una foto per me?

Do these sentences translate as "Can you take a photo of me?":

Puoi farmi una foto?
Is "farmi" a combination "fare" and "mi", and not a tense of "fare"?

Puoi fare una foto di me?

Grazie mille per tua risposta

Hi JackK15, 

Thanks for your question!

The sentence "Mi può fare una foto?" translates as "Can you take a photo of me?" - it's important to note that the sentence addresses someone formally as it uses the formal You (Lei) instead of the regular informal you (tu), which would otherwise be "Mi puoi fare una foto?".

The sentence "Puoi farmi una foto?" translates similarly as "Can you take a photo of me?". In this case the sentence is informal (informal you). If you wanted to make it formal you should say "Può farmi una foto?".

As you can notice, "farmi" is the combination of "fare" and "mi" (which means "a me"), as you correctly stated, so the meaning won't change.

Finally, the sentence "Puoi fare una foto per me?" would mean "Can you take a photo for me?", perhaps used in a context where you are unable to take a photo and you ask someone else to do it for you. In this case the sentence is informal (informal you). If you wanted to make it formal you should use "Può fare una foto per me?".

Hope this helps!

Thanks for your response Caterina. 

Are "Mi puoi fare" and "Puoi farmi" used interchangeably?

Indeed, they are interchangeable!  :)

eg. "Mi puoi fare un piacere?"  (ENG: Can you do me a favour?)  is the same as "Puoi farmi un piacere?".

Thank you. It's good to know that piacere can be both a noun and a verb! In English, favor  can also be both a noun and a verb. 

...Exactly! 
A synonym of "piacere" (noun) would be "favore", just in case you wish to use both of them :)