"No, thanks!" should be considered correct, even though it is not very formal.
no thanks is less formal and you would use merci, it is the difference between thanks and thank you
Yes! Mulțumesc is the romanian word. Coming from French you can also find ”mersi” (with s) but this is regarded as an ”imported” word and as such some people loves it while others disregard it.
Maybe I'm finding too many false cognates, but the part of the word "Mult," suggests being derived from many "multi"... so I'm wondering if Many thanks. or Thank you very much isn't also an adequate translation here!
"mult" does mean "much" in romanian, however, the word "multumesc", even though probably derived initially from mult+something, is the only word for "thank you" (except the borrowed "merci/mersi"). There is no lesser form of thank you below Multumesc; to say thank you very much, we say "multumesc MULT" (or "multumesc foarte mult", foarte = very).