Solutions to problem 1:
Wash your hands before playing. Try using a string lubricant like Fast Fret. Give Elixir strings a try.
Solutions to problem 2:
Make sure your strings are wound neatly around the
tuning pegs. Wrap once above the hole and then 3 or 4 wraps below the hole, going down down the post towards the headstock face. Don't overlap your wraps. Only use as much string as you need.
Does your guitar have a trem or a
fixed bridge? Locking or non-locking trem?