Leave your betta in there if you aren't going to do too large of a water changes. In my 5 gallon, it gets as low as 3-4inches, including gravel. That's about 2-3 inches of swim space for them (in my larger changes). They do just fine as long as they can swim upright without dragging against gravel.
In your 2.5, I would recommend a siphon/gravel vacuum/gravel cleaner. (All the same things). It's much easier to use, but I would get a self-starter, or one with a pump. I use the bi-orb one, it's very simple. Your betta will get used to it.
When you put water in, you finger-test it. You compare the fish-tank temperature, and MAKE the tap-water temperature match as close as possible. After doing this for a while, you will get used to it. Also, I pour my tap(with chlorine) water into a separate bowl, treat it, and then put it into my tank. However I only do this because I don't want to affect my cycle. I think if you're uncycled, it will be okay to put it directly in.
The Tetra heaters (2-15) seem pretty good for smaller aquariums. I'm not sure about the comparison to the mini hydor however... I'm curious about those two.