I know that you have already bought your silk plants, however your fish will appreciate live plants more. Broad leafed plants like Anubias Nana, or low light plants such as Java Fern, Water Wisteria, Water Sprite, etc are all much better choices for your tank than what you have now. Over time that silk can accumulate algae and become quite unsightly requiring cleaning on a semi-regular basis.
That said I'm sure the betta would like that set up you have now much more than a cup or a bare tank. He may appreciate the silk plants but he'll appreciate live even more.
If I did buy some real plants, could I have them AND silk plants?
Would I have to cycle the tank before I put the plants in?
Do they need a specific type of substrate? I currently have white(ish) aquarium gravel.
You can never have too many plants! LOL, that's probably not true, but I like the well-planted look myself as well.
I'm in the same boat as you are about the live plants, never tried it, but interested. I've had rec'd that I start with floating plants, duck weed and things like that, because they're easy, grow quickly, and they can get air from the surface. :) I haven't started yet, bought some bulbs yesterday and they're soaking/working on growing right now.
Oh, and my plants touch my filters and heaters all the time, no problem.
Taking on live plants can be a very large project. You need to decide what plants would go best in your tank, find the right substrate and probably get fertilizer for them, maybe even CO2. You'll also need to set up the right lighting for your plants. There are a lot of good Betta plants, but do some research before trying them or you'll just end of with dead plants and dirty water.
Of course, silk plants are perfectly find. Betta use plants for cover, not for food, so he's not really going to mind which is in there, it's personal preference and how much time, money and effort you want to put into your tank. Yes, you can mix silk and live, or if you want to just put a moss ball in there with all your silk plants, that's fine, too. It won't do any damage.
If you're going to place driftwood in the tank, remember it will seep tannins into the water and might discolor it. While this is great for the fish, some owners don't like the look. There's also fake driftwood, right? And rocks.
I wouldn't bother with the filter, but if you're going to, don't worry about anything touching it, but try to keep objects from being sucked into the intake. If you need to place a screen or something over the intake to keep things out, that's fine.
You can let your plants touch the heater, it won't get hot enough to melt anything, if that's what you're worried about. Almost all of my betta love sleeping on their heaters, they're safe :)
It looks like a nice little tank, give it some light and maybe a cave of some kind down there in the middle and you will have a very happy fishy.