1. How much light do I need for the plants to carpet/spread? I currently have a GE Aqua Fresh and Saltwater 15W bulb and I was wondering if that is enough for the plants to spread?
The light is on the low side for the hairgrass and HC but I think may be OK, however, I think if you add CO2 to the tank you'll have better results.
2. Is the filter really needed, I heard the current from filter can be dangerous to a Betta.
Yes, I think a filter is a good idea. I would recommend putting a sponge over the intake for the filter both to kep the betta's fins out of it and also to keep the shrimp from being sucked in, however. Bettas don't like FAST flow, but some flow is fine, and actually very good for planted tanks.
3. Do any of these plants need any fertilization, and if so what is the cheapest? Tablets? Any brand specifically?
You can probably make do without ferts, given your lighting and bioload. Seachem makes a good series of liquid ferts, but dry ferts are the cheapest way to go over the long run (though there isn't nearly as much savings with a tank as small as yours, so I'd probably stick with the Seachem line). You can use root tabs if you'd like.
4. Can the Dwarf HC, Hairgrass, and Sword spread into a carpet without fertilization?
They'd all probably spread much better/more quickly with ferts and CO2, though the chain sword will probably do the best out of the 3 in your current setup.
5. Now that there is a lot more hiding spaces can my Betta live with a cherry shrimp and an otto?
It really depends on your betta. Some will hunt and eat shrimp very aggressively, others ignore them. The oto is probably just fine.
6. I just ordered a nano CO2 diffuser, will that be enough to keep these plants alive?
Probably, since you've got a small tank. You'll just need to make sure to keep your DIY CO2 mix nice and fresh.