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.