Be aware that the Mini inputs are Line level.
Midi control is built in.
Open Audio Midi Setup in Applications>Utilities, see the input & output options & KHz setting there.
Tough to say on the 8 GB, but I'd call that minimum today, but you can add your own...
In addition to BDAqua:
You will probably want to purchase and use an external "sound card" or DAC via USB or FireWire to get decent audio quality from external sources into the Mac.
Audacity is sort of an 'industry standard', but not for complex recording purposes as I read in (and in between) your lines. Free on bord with a new Mac will be Garageband. Once you've got your head around it, you'll love it.
Gaming - I have no clue, but don't expect the lower-end models of Macs ideal for heavy gaming... even the higher-end versions can lack, depending on the games. Check around here and elsewhere, there are gazillions of gamer forums.
Budget & portability: Keep in mind that for a Mini you'll need an external screen, and if that's supposed to be your TV, make sure it accepts data (PC or Mac) signals on HDMI. Or has DVI as well. Otherwise disappointment - not due to Apple!
Eventually take a look at MacBook Pros, they are portable, do come with a screen, and can do more than an Air - depending on model.