If you click on the link in my sig, you'll see a very happy betta in 3.5 gallons with lots and lots of plants (and even more plants now!) and five red cherry shrimp. The ammonia's nil. I do two water changes a week, one small (20-25%), one with a gravel vac (50%). I never cycled the tank on purpose, and the ammonia has never got past 0.25, ever - I think the plants help with that. I also never do 100% changes and there's parts of the tank with rooted plants and stuff that I can't vacuum at all.
I did restart the tank after my sickly-from-the-store lovely Sid passed away, and used Seachem Stability for a couple of weeks, not so much to kick off a cycle, but to stabilise the tank. That seemed to work pretty well. I've had no health problems at all, and the fish, shrimp and plants are thriving.
So yeah, smaller tanks can really work. But here, you've got a 6 gallon! Lucky you. :)
You don't have to cycle the tank at all, if you're prepared to make a couple of 50% water changes every week and gravel vac once a week - if you have a lot of low-light live plants like wisteria and anubias, etc, and a decent light for them (6500Kelvin) you can do less changes as the plants utilise fish waste to grow.
If you want to cycle it, it can take a few weeks. I'd highly recommend leaving the fish in his bowl for that process. There's a lot of good info on cycling in the stickies here, I've read them a lot so I know what to do to cycle a bigger tank I plan on... I've read that gravel is better for cycling. Easier to vacuum, too, than pebbles, I ended up taking all mine out and using for them for 3 gallon unplanted my daughter's plakat is in, since it gets a good wash out once a week anyway.
Can you take the tank outside and hose it? That might help with the grime. A brand new toothbrush or even a new, clean mascara brush (the kind you get without mascara with it, lol) is good for tiny corners and tricky bits. If you use bleach, rinse it well (give it the hose!) and leave it in the sun a few days to deactivate the bleach. If it was a store tank, I'd probably do that, who knows what was in there..
And hey, if your gf can't have the tank after all that, consider getting her something like mine, which doesn't take up a lot of room but has plenty for the fish to swim about in (unless it's HUGE like a giant..). And maybe keep the 6 gallon for yourself! Fish keeping seems like a stressheadache nightmare at first as there's so much to learn all at once, but wow, then it all kicks into place and the enjoyment is all really worth it.
Then the addiction kicks in..
ETA: also, this rapid swimming.. I hate to add to your stress.. but if he's smashing himself against stuff, he could have ick. Check for white spots that look like grains of salt. Or velvet.. shine a torch on him, see if he shines really gold. Some fish do spazz out in the tank anyway (my girl does it all the time, the dork) but if he's really hitting stuff fast, it could be him trying to scratch himself.
Ick and velvet are both pretty easily and quickly cured, so don't panic if it's either of those!