Small tanks don't cycle........you have to do 2-50% water changes a week for one gallon tanks. Filters aren't necessary as long as you keep up on the regular maintenance. You do need to keep the tank in an area where the temps do not fluctuate too much and where
I have some one gallon betta tanks that do quite well actually. Here is my "system" for keeping them clean....
Twice a week I syphon out half the water and replace it with clean aged dechlorinated water. At least once a month, but usually more often, I cup the betta and dump everything in a colendar/strainer....rinsing out the gravel using the old water, rinse and wipe down the inside of the tank and clean up any plants/decorations I have in there as well and then replace everything with 100% clean water.
I have one planted one gallon that all I do is syphon out half the water regularly....twice a week.......and once a month or so syphon out almost all the water, but not all. Because of the substrate and plants I can't and actually don't need....to clean off anything in the bottom, though I do occassionally wipe down the sides. This tank is almost trouble free...the live plants have alot to do with that I believe. I have one water sprite and a small amazon sword in there.
You can keep one betta and a shrimp or two in a tank that small. The bettas may end up eating the shrimp eventually, but I have a betta in a 2 gallon with a shrimp that's been in there for quite some time now. I don't even know if the betta realizes it's in there !
YOu have to be very good about keeping the water and the tank clean if you are going to put any fish or anything in there Crazy.......I enjoy cleaning out my tanks so it's not an awful chore for me. If you think you will be good about keeping up with it, I'd say try one betta OR two ADFs with a few live plants. Java fern or water sprite are a nice looking plant and very easy to keep.