Okay. You need to be sure you have a heater. The thing that makes the water warm. That isn't the filter, it's a separate thing.
Your spotty fish has ich. This is a parasite - don't panic! - easily gotten rid of in most cases by upping the tank temparature, and using a dose of Aquarium salt (not regular salt) at 1 teaspoon per gallon, with 100% water changes every day.
You really do need another tank for the other guy. He will very likely also have ich, it's highly contagious. You'll need both tanks heated. A filter isn't as necessary as a heater (adjustable is WAY better since you'll need to change the temp..). So go to the store, get a tank and a heater for that guy because treating both fish in your set up, and them escaping etc, is going to be a nightmare, probably.
Both fish will need their temp --slowly--- turned up to 86, the spots will fall off. Every day you have to change the water and add new salt, for about a week. The ich should go away and not come back. Then --slowly-- turn the temp back down after a last 100% change. This will all be a lot easier if you don't have gravel in the tanks. If you do, wash the gravel out when you change the water.
Their behaviour is probably a mix of stress/being new/not being acclimated (slowly introduced to new water/ph)/having ich etc.
They'll be just fine once all their needs are met.
The hanging out at the top of the water is normal, if they're just going up, taking a breath and swimming off. Bettas breath air.. they have a special organ that helps them do that, and they can drown if they can't take a breath.
Bettas also jump, so make sure your tanks have a lid.
In a 2.5 filtered tank, 1 30-50% change plus 1 x 50% change with gravel cleaning (a vacuum is handy for that) per week is fine for one fish.
When putting fish in a new environment, float them in the little store cup and slowly add new water to it every 5 mins for 15 mins or so. This helps avoid shock.
Do not add old tank water to new if the fish have had illness/parasites.
And really -- don't panic! It's not that bad. We're here to help you. This happened to me with my very first fish (the ich, tank not right, etc) and now I can be calm about things, lol.