Equip card question

Two questions:
1) If I have an equip card on a monster and the guy flips over Penguin Solider, does my monster AND the equip card go back to my hand or just the monster?

2)Two part question: If my monster is Set, and they use a card effect to take control of my monster, are they allowed to since the card is Set? Second part of the question is when they equipped my card to their monster card(they were allowed to do that), are they allowed to just detach it whenever they want or does the monster itself have to be destroyed? Reason being, I had spatial collapse on the field and he was at his 5 card limit. He just wanted to detach the card so he could summon another card, was he allowed to do that?

1. equip goes to grave

2. mc works on face down monsters
no they cant detach it whenever they want

1)Unless he targets the Equip also, no.

2)Face-Down monsters cannot be targeted by equip spells.
As for the second part, no he cannot just send a card for no reason, something has to get rid of it.

and penguin soldier can only target monster

1. equip goes to grave, monster goes to hand
2. Face-Down monsters cannot be targeted by equip spells.
As for the second part, no he cannot just send a card for no reason, something has to get rid of it.