The algae popping up may be as a result of too much light, your plants not "outcompeting" the algae for nutrients due to deficiencies of their own, low Co2 levels. Or a combo...
My guess would be low Co2 level and too much light. How many hours a day are your lights on and what type of lighting do you have? (Watts per gallon?)
Co2 levels are low unless you are adding it to the tank.
DIY is a cheap way to start Co2
Pressurized is much better but also comes with a hefty price tag. Seachems "Flourish Excel" can be used as a Co2 substitute. It is a liquid form of carbon. I use this in my tank currently and it works well.
SNAILS!!!!ACKK....my advice keep a very close eye on the tank and remove any of them you find. Unless of course you want thm in your tank. Mine started off the same way and now I pull out 10-30 per day. I will never get rid of them without breaking down the tank. I should have been more careful with dipping new plants.
As far as trimming goes....cut the stem just above a leaf. You can also replant the old stem as you have pointed out already. Tank sounds nice...any pics?
Best Wishes,
Nate