Well, I'm rather a newbie too but I find that my brain hangs onto things better when I try to explain to others so here goes.
Do the bacteria in the water convert the nitrite to nitrate? I know nitrites turn to nitrates but I don't believe it's the bacteria that does causes it. I hope someone will come on and explain that better because I know the cycle but not the "why" of it.
Is the nitrate harmful to the fish? Some nitrates are acceptable, I believe under 20 is OK and anything over 40 is the "danger zone".
Should I get a Nitrate test kit? Yep, I think you'll find it incredibly helpful for when you are cycling your tank and so you will know if you have problems.
Should I have real plants?(I don't like them so far because the leaves rot and decay) Real plants can help but obviously not if they are just rotting. If they are rotting there is a reason (like not enough light etc). Plants do better in a cycled tank as they really like the nitrates - which is why they can be good. I would try to figure out why your plants are failing and reconsider as I also think they look a lot better too.
Do I "need" real plants? Not absolutely necessary but see above.
And what is the max number of fish for my tank size? The general rule of thumb is about 1" per gallon but that is REALLY basic and depends on the kind of fish. For a while I had two ten gallon tanks, one coldwater and one tropical. I had six neon tetras, six pygmy corys and a dwarf gourami in my ten gallon tropical and was maxed out. I had only one fantail goldfish in the other ten gallon and was maxed out. So consider what type of fish you want, how big they will grow, whether or not they prefer to school/shoal and whether they are big waste producers (like goldfish). So research, research, research and come here and ask questions, people are great!