Hi and welcome to the forum, coet. Yup, we love people like you because you want to give your betta a better living environment and that makes us happy.
First, the UG heater. Is it the Zoo Med Betta Therm heater? Little round black thing? If so, it will heat the water 4-6* degrees over the room temp. But I don't think it has any kind of auto shutoff once it reaches that target temp so you'll have to use the thermometer to monitor and turn it off manually. Once you get a larger tank, I'd suggest getting a different heater that is both adjustable and turns itself off when the set temp is reached. 81-82* is on the higher side of a betta's comfort zone which is about 76-84* with 78-80* being ideal. If your room where you keep the betta stays pretty warm, you might be able to get away without a heater until you upgrade. It depends on whether or not you want to keep plugging/unplugging your heater.
Omega One pellets are way superior to Hikari now. I used to feed Hikari and granted my bettas loved them but Hikari switched its formula to have more wheat meal and less fish product so it's not as nutritious as it used to be. Too much wheat product can lead to constipation. I'd stick with the Omega pellets.
Berry will be okay in the 1g tank for three weeks as long as you do a 100% water change every other day. In that small of a tank, the ammonia will build up pretty fast. Prime will help but you still need to change the water pretty frequently. But considering he was in a cup before, 1g probably seems like a whole ocean to him.
I had the Marineland Crescent 3 for a short time but I did end up returning it. I found the lid had some trouble staying on and I didn't care for the filter. But that's just me so you may find you love it and it works perfectly for you (especially if you get it on sale
). It is a nice looking tank and has a pretty small footprint so it's a good tank for a betta. I also really liked the LED light fixture of the Marineland Crescent. I don't know much about the Aquascene aquarium kit so I can't advise you there.
The lid of the Marineland Crescent does have a bit of a hole but unless Berry gets really spooked and happens to be right under it, I'd say the odds of him jumping out are slim. If you're not sure, you can always cover it with something, like a book or tape some paper over it.
Since you just got Berry yesterday (probably two days ago by the time I post this), then it's normal for Berry to not be hungry yet. It can take a betta a week to settle in and start eating. Plus, it sounds like he's a little shy still. One of my bettas was so shy I was afraid she wouldn't even come up to the surface to breathe let alone eat. Try putting his pellet in there and just standing back a distance so you can see but you're not hovering over the tank. He may be brave enough to grab it then. If he hasn't eaten after ten minutes, go ahead and remove the food though so it doesn't dirty the water. Also, just getting Berry used to you will help. Spend as much time around his tank as possible, sometimes just resting a hand near it or putting a finger to the wall (don't tap) and leaving it there. He might swim away at first but if you leave your finger or hand on the tank wall long enough, he will come back to investigate. Also putting your hand over the tank and just leaving it there for a few moments is a good way for him to realize the hand is harmless. And before you know it, he'll be eating and begging for more because bettas are little piggies.
Most people here can't agree about how often to feed freeze-dried foods. I feed my bettas one or two freeze-dried bloodworms (small pieces) just about every day and have no problems. Other people say the food expands and causes bloat and constipation. I'd say feed what you are comfortable with but never more than one or two small less than a quarter inch long pieces at a time and never for every meal.
If you have any more questions don't hesitate to post. Everyone here loves to help.
Good luck with Berry!