No need to say sorry for asking good questions here!
With the fish you have in mind, i'd really suggest keep measuring your water every day when you get your kit, one day ammonia will rise (and then the NO's will follow) and then fall again (which may happen in 1 week like you wish, but it may take a few days longer), after that point is when I'd add fish in there, preferably I'd start with the Betta and then another week later the Neons. That's really the safest approach to avoid loosing any one.
Snails are EXCELLENT lil helpers in tanks to keep it clean with you, so pers I prefer having some in all of my tanks, so clearly my vote goes: Keep them
And no they can not cause no problems neither to live plants (if you wanna have some) nor the fish.