If I miss any of your questions please let me know! Lack of sleep has my brain a bit fuzzy.
No, they do not need a bubbler, or technically a filter. Both of those will oxygenate the water, and the filter will additionally keep the water clean, BUT betta have what's known as an labyrinth organ which means that they can take oxygen from the surface of the water to breath if they are in low oxygen water. Contrary to rumor to do not need to use that organ, and if they are in water with a high oxygen you won't see them come to the surface to take a gulp of air as often. My own 2.5 gal tank has neither a filter or bubbler right now and hasn't since I got the tank. Some betta's like to play in the bubbles, some don't care either way, and some don't like them at all so it's just a matter of preference for them. If you wan't to take the bubbler out the bottom of the tank feel free! You can always put it back in later when you have something to weigh it down.
I don't know if you have a filter on the tank or not, but if you do not you will need to keep up with the water changes because the ammonia and nitrite will rise even faster. Also an unfiltered tank is very hard to cycle due to the beneficial bacteria having less to grow on. The bacteria likes to grow in the filter sponge and to a much lesser degree it'll grow in the gravel and on the tanks decorations.
I clean my API test tubes, and the caps to the tubes, by rinsing them really well in hot tap water then turning them upside down to dry.
Here's the forums guide that explains how to do a fish in cycle... https://www.bettafish.com/30-betta-fi...-tutorial.html
It looks like the ammonia test you did shows .25 ppm . Am I reading that right or is the color off due to it being a picture. If it is .25 did you test the new tank, or the old tank? If it's the new tank was the water completely fresh? The reason for all the questions is that if you tested new water, then it should have 0 ammonia, but sometimes tap water contains ammonium (not toxic to fish) and the API test will read it as ammonia. All that to say if it's new water do a test on your tap water to double check that that's the problem. It'll save you a bunch of headache later on when you are cycling the tank.
When testing the tank water while using Prime make sure to test the tank water right before a water change, or 48 hours after a water change.
A couple of things with the heater, it may be that the thermometer is not working correctly, I've had that happen. If the thermometer is right then try unplugging the heater for 2 to 3 hours, leaving it in the tank, and then plugging it back in. If it's still not heating the water then you'll know it's the heater that's not working correctly.
Hopefully that answers all your questions, if I missed any just let me know.