Here's a sort of popular vote by internet usage: Google hits for "rightdoing" = 75,900 as of the time of this post. Notable usages include a published book title, a more modern published article, and a quoteworthy quote. Can't find it in any dictionaries myself (besides urbandictionary!), but that doesn't mean a native speaker like me would see any reason to object. (In fact, I wouldn't!)

P.S. I should add that I can't compare the hits for "right doing" because they might appear in the middle of a larger sentence like, "I don't feel right doing my comparisons like that," in which case it's not just one noun; doing becomes a separate verb.

In British English, the correct phrase is hand in.

From Macmillan dictionary:

hand in

phrasal verb (transitive)

1. to give something to a person in authority
   Please hand in your keys when you leave the hotel.
   All essays must be handed in by Tuesday.

In American English, you can use either hand in, or turn in:

turn in

phrasal verb

ᴍᴀɪɴʟʏ ᴀᴍᴇʀɪᴄᴀɴ

4. to give something to the person who has officially asked for it or who is in charge
   turn in something to someone: Mitro turned in his resignation to the President.

There is a subtle difference between the two - you can turn something in only if it has been specifically asked for (you wouldn't normally talk about turning in your keys as in the "hand in" example, unless perhaps you mistakenly ran off with them and needed to turn them in later) or you are giving the thing to a superior.

So, for example:

• Please turn in all borrowed equipment
• When I found a stranger's wallet on the street, I handed it in
• I have done my homework, and turned it in