Welcome to the forum!
The only filter I will use with bettas anymore is a sponge filter. They use bubbles to push water upward through a sponge, and the good bacteria grow on the sponge. They create the lightest current of any filter. I can't recommend any power filters for that size tank as I've never used any.
If you stay with the power filter, I would forgo the carbon. Cycles are more difficult to keep in a smaller tank, and since carbon needs to be changed every week in order to remain effective, you would be removing a good chunk of your biological filter on a weekly basis. Cram that sucker full of bio-media like ceramics. The more surface area your filter has, the more stable your cycle will be.
In order to prevent your betta from being sucked up in the power filter, some people will wrap an aquarium sponge around the intake. It has the added benefit of being more surface area for the good bacteria as well as preventing your betta from getting caught.
If by old-fashioned thermometers you mean glass, then I would go for the glass over a digital one. Good digital thermometers are more expensive than the glass ones. And yes; they work just fine. I have them in all of my tanks ranging from 1 gal up to 55 gallons.