You probably could, but I don't know if your Yamaha has a phono input. Search for "RIAA curve" if you want to know more, but to keep it simple, amplifying a turntable signal is not just boosting the volume, you need a specific equalizer for it.
There are also digital equalizers for this, btw. So if your M-Audio input is sensitive enough to pick up the phono signal without adding too much noise, you can perform the equalization together with the general cleaning. There are various RIAA VST plugins, and Audacity can perform RIAA Equalization as well.
edit: regarding the software: according to good hifi tradition, if it ain't expensive, it can't possibly be good enough, right?
For serious recording I would suggest any recording software that understands ASIO, so Cubase qualifies. It also supports VSTs, so you could even do the equalization there. Don't worry about the file format, you're going to keep it in Cubase Project format anyway until you're done editing. Then you can choose any format you like: aiff, alac, flac, mp3...