Growth will depend upon light, ferts, CO2, and how much you are starting with.
Give them enough light and some ferts, I use Seachem Flourish, and most plants grow quite well. Adding CO2 helps boost growth, but if not going high-tech, or using mosses, thwy aren't absolutely nessecary.
Dwarf/pygmy chain sword is a good fast grower that works as a carpet.
I have about 4.5 watts per gallon I also have Diy Co2.
How long should it take for this tank to grow?
Also when I'm planting the carpet plant should I be planting stem by stem or plant the plant with the small pot?
If it's in a pot, remove it from the pot. Remove anything, rockwool, gel, et cetera from around the roots and pop em in.
4.5 watts a gallon? Wow on a 10gal., mine are only 2 on my 10 and 55. But I don't have any high needs plants.
Have not gone the CO2 route. Just don't forget the ferts and it should grow fairly quickly.
Lights, nutrients, CO2. When one of these gets out of whack algae gets going.
Which type of hair algae? Different types need different methods for eradication.
A description of both or pictures would help.
First algae is normal hair algae. The other algae growing on the glass is whitish and stringy? I'm going to add plants a dwarf puffer and a clean up crew of 2 amano shrimp and nerite snails.