1 gal pickle jar!

very pretty fish love the color..and the jar looks pretty cool too...

a 1 gallon is do-able if you keep up on the water changes, but i wouldn't recommend it. and when using small enclosures, wide and long is better than tall and skinny - gives the betta more swim room in a wide/long.

if you are set on keeping him in there, then daily water changes have to happen, and at the very least 50% (i'd recommend more than that - closer to 75%). and keep an eye on ammonia (test frequently) so that you know you are changing out enough water to bring the ammonia reading back to zero every time.

as for filtration, a sponge filter or internal filter is going to take up A LOT of space, and a hob filter will make a whirlpool of the jar IMO, even if you can find one to fit the rim of the jar.

and leave the mopani out - it'll take up a lot of swimming space too.

I would get him a little 2.5gal of larger tank with a little elite filter and heater. I'm sure he's happier in the jar then being in a cup but you could make him happier

bettas (and all other tropical fish) do not handle temperature fluctuations well at all.

good idea to put him in a tank if your house tends to go between warm and cool.

i wouldn't put any critters in that jar if the temp fluctuates like you said.

Ghost shrimp would be ok if you could get a 7.5 watt bowl heater. I think msjinkzd was selling one (used) for pretty cheap, they are not too expensive.

A betta would be so much happier in a 6 gallon tank, if that is an option I would definitely move him. They sell hexagon undergravel filters, that would probably be your most discreet filtration option:
http://www.bigalsonline.com/productDetailsPage.mtw?productId=9340161

If it's too big, I'm pretty sure you could cut it if you're handy and have the right tools!

The only fauna I'd put in a 1g unfiltered, unheated jar (if I put anything in at all) would be a couple of pond snails. Betta splendens (what you have) thrives in temps. ranging from about 75-80, maybe a little higher and enjoy gentle water movement; I once read in a husbandry book that bettas' muscles can become atrophied over time when in a setup lacking any current. When I first started keeping fish a few years ago, I tried a betta in a setup similar to yours and the betta didn't make it past a few months.
Since then, I have learned to properly house them with heat and mild filtration. If you use filtration in your betta setup, go with a sponge filter if possible, as hang-on-back filters and even submersible filters provide too strong of a current for betta splendens with long, flowing fins.