Most craft stores should have origami paper, Hobby Lobby usually has it, but just try asking at your local craft store.You could order it online, but im not sure where to physically buy it at.
the more complex, the bigger paper you should have. it won't get too thick if you use big paper. for example, the origami bahamut, or divine dragon was invented with HUGE paper, by kamiya satoshi
That depends on the brand and type.Most paper is...just paper. Usually coloured on a single side, or both sides different colours.You can get origami paper made from Japanese Washi, but it's more...