A card you 'control' is something on your side of the field, be it a monster, spell or trap, and it can be something the opponent owns, but you've taken control of by whatever means.
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To promote cards you need to use them in Duels. Have them face-up during the Duel. Have them defeat other monsters and have them attack Deck Leaders. The more you use a monster, the more likely it is that it will be promoted. Note, however, that only monsters can be promoted, not spells or traps.
Yes, it will be unaffected by the effects of any and all types of spells, whether they be Normal, Quick, Field, Continuous or Equip. The Legendary Fisherman for example gains protective effects if Umi is in play, however it therefore becomes immune to Umi's effect and doesn't gain the ATK bonus it would normally grant.
The rainbow foil cards are rarest, then the foil cards.
yes you can equip spell card cards to an opponent's monster unless the card states so a good example of this is snatch steal