First off, welcome to the forum!
1) 2.5 gallons is typically considered the minimum. Some people keep their betta in smaller tanks for different reasons, while other people say that you should house a betta in no less than 5 gallons. I personally have all of mine in 5 gallons, but that's primarily because I find it easier to maintain a tank of that size or larger. While a female or short-finned male may appreciate more space, a heavier-finned male could do perfectly fine in a 3.5 if that's what you'd like to do. At the same time, a 10 gallon tank would make a typical female or short-finned male very happy, could house more tankmates, and could take less maintenence.
2) Low light plants could work with LED lights - some good ones are Marimo moss balls, Anubias, Java Fern, and Java Moss would be good options. They grow more slowly, but are pretty darn hard to kill.
3) There aren't any plants that I know of that you shouldn't use for bettas, as long as the plant is fully aquatic. All the plants I mentioned in #2 are fully aquatic and do well.
4) Don't mix males and females, even if you intend to breed. 4+ females can be housed together in a sorority in a tank that's at least 10 gallons, with a 20 gallon long being a good size, that is very heavily planted and has lots of places to hide. From what I've heard, a sorority isn't something I'd do for my first time having a betta. But if you went with a 10 gallon, it is possible to have tankmates.
5) Bettas can get bored and benefit from some sort of stimulation. This can be in the form of having a well planted/decorated tank for the fish to explore and patrol, to "playing" with the betta, to having tankmates. I personally don't think I'd keep other fish with my betta, if only for the fact that you pretty much need at least 10 gallons to have other fish, I don't think that many fish that could
live with a betta are well suited to a 10 gallon, and there's always the risk that the betta may be stressed by the other fish or vice versa. That said, I have pond snails and malasian trumpet snails in my tanks and one of my bettas lives with some ghost shrimp with no problems.
6) Pretty much covered it in #5 - a lot of people like nerite snails, and I have a few types of "pest" snails and some ghost shrimp. Some bettas will tolerate shrimp, others won't. I'm not fond of the idea of having an animal solely to clean because I don't think there's much net gain - yes, they may eat some of the algae and/or waste products, but they also poop it out. Most snail species kept in the aquarium poop A LOT, as do pleco species (and there aren't many of those that do well in smaller tanks, anyway).
7) As long as you're taking filter media from a healthy tank, it's no problem. Cycling is the same concept, no matter the species kept
8) In a small tank, you don't always need a filter unless you want to cycle the tank, but then you need to do more water changes. If you do have a filter, you don't have to downsize the filter size, but it's usually a good idea to baffle the filter to reduce the flow. Many people use the Aquaclear 20 for their 5.5 gallon tanks with no problem. I put a piece of filter sponge around the intake of my filters to keep their fins from getting cought (this also works as a prefilter and provides more surface area for beneficial bacteria) and use a water bottle to baffle the flow going out. This can also depend on the individual betta - some can deal with a small current better than others. I have one betta that's a very good swimmer and would flit around the tank with no problems and another that would be pushed around by the smallest current.
EDIT: Good grief, I need to take less time writing replies (and try not to get sidetracked while doing so)... Sorry for repeating what everybody else has said!