Hey! The first plant looks like an anubias, and the 2nd one looks like a java fern. The first one is just getting dust algae on it and the 2nd one just has a pottasium deficiency. I wouldn't worry. It's fine and both plants are ok.
Well, the cucumber is just an extra. They mostly live off algae so you need that in your tank. There's not a lot of algae in my tank so they also eat uneaten fish food.
Tough also dropping in an algae wafer weekly would be cool if you wanted to be sure. I don't have nerites, I only have bladders...
As long as there's not too many snails, you don't need to give them calcium. I don't give calcium to mine.
Before you add them, test your betta's temperament with a bladder/ramshorn/pond snail. They are great test buddies.
Your stockking ideas seem nice. I'd add the gourami last to reduce possible aggression. And did you say "mix of neocaridina"? Don't put diff species of neocaridina together because they will breed and the babies will sort of brownish.
For the centerpiece fish you can try a ram. It's one of the fish I am also going to get soon, and I've heard they are peaceful. They're really colorful and they seem like very fun and active fish.
My vote goes for ember tetras. I have 8 of them atm. They are gorgeous fish. Tough they can be shy...