The flame -2, regen etc is also for the opponent right? Then, is the option to pay to remove corrode and burn also for the enemy?
Whenever a card's text vaguely mentions "you", assume it is talking to the controller of the card. So, when [mwcard=MWBG1J04]Raincloud[/mwcard] says "you may remove 1 Burn or Corrode" it means that the controller of Raincloud may remove 1 Burn or Corrode. Neither your opponents nor your allies receive this ability, just like they are unable to pay mana to move the Raincloud.
But for the sake of clarity, the first line of text does apply to the opponent's cards since they are objects
You are correct. You're opponent's objects also gets Flame -2, Acid -2, and Regenerate 1