Fluval edge, (have not had one, they look like a pain to maintain considering their small opening) would be a aesthetically pleasing 6 gallon tank. I have seen lots of tanks at petco/petsmart that would fit in for what you need.
Is the tank going to be heated?
Depending on what you get you could easily get by with a sponge filter which would mean that you would only have to look for a silent air pump and/or enough tubing to get the air from the pump to the filter.
For fish, small tanks tend to be harder then larger ones and a lot of the small fish seem to be somewhat more difficult. Are you planning on a planted tank? If so something you might like (as long as your tank is 5+ gallons) is a pea puffer. You would only be able to have one and maybe some shrimp but from what i have read that have pretty big personalities for small fish. They dont need the teeth trimming other puffers need either so you could basically feed em any frozen/freeze dried food you could get em to eat or theirs always the live snails that they enjoy hunting.
Theres also Scarlet Badis
which is a micro predator but a very pretty small fish. they stay under an inch long though. You could do 1 male and if you could get your hands on females a couple females but females are hard to find. I've gotten mine to eat frozen brine shrimp as well as whatever is growing on the plants thats to small for me to see. Another fish with 5 gallon min. 5 gallons per male and up to 1 male to 3 female ratio.
I don't know much about them but i have read that sparkling gouramis would do well in smaller tanks around the 5 gallon range as well.
Theres probably a lot more fish out there that would work i just don't know them. Id say a safe bet though would be to go planted. Helps keep the water clear and the aquarium stats in a safe area. If you go with something like the fish i mentioned you should check out dwarf shrimp to go with them. Stuff like Snowball/ Red Cherry/ Pearl blue shrimp to add some more color and life to the tank. If you let them establish before you add fish in there they should out breed even mild predation.