#losing abilities

Mutate isn't really as complicated as most people think it is. A mutated creature is basically just the top card of the mutate stack with all of the abilities of the other cards in the stack. Just add their text boxes on to the top card's text box and that's what the creature is.

The abilities of all the cards that make up a mutated creature are consdered to be its printed abilities, so a card like Reprobation will remove all of them, no matter when they were added to the mutated creature. (A mutated creature's timestamp is the same as that of the earliest card in it. Every other card merged with that permanent and so no new timestamp was created.)

That interaction is working as intended.

While it’s true that effects that add and remove abilities apply in timestamp order, mutate doesn’t work by giving abilities and the like, but rather by changing the copiable characteristics of the mutated creature. Since mutate is a copy effect, it applies earlier than Frogify’s effect and so any ability that would be gained by the creature is stripped away by Frogify before it can be relevant.

Whether or not your Army has flying depends on which effect is newer. If Transmutation entered the battlefield after Skylord did, then your army doesn’t fly. If Skylord entered the battlefield after Transmutation became attached to your Army, then it does fly. 

A double-faced card that transforms without leaving the battlefield, which is most of them, is still the same object it was before it transformed except with different characteristics.This means that when Thing in the Ice transforms in your example, Awoken Horror won't have any abilities and no creatures will be returned to their owners' hands.

Tarmogoyf will be 0/1 until end of turn.

When Merfolk Trickster’s ability resolves, it makes takes away Tarmogoyf’s abilties. This means that it no longer has anything defining the *’s in its */1+* P/T. Since the * is undefined, 0 is used instead of *, making Tarmogoyf 0/1.

Yes, Randy can still cast Treasure Map here.

The ability to cast the card exiled by Hostage taker is part of the resolution of Hostage Taker’s ability, as is the limited duration of the exile. If Hostage Taker loses its abilities but remains on the battlefield, the exiled card can still be cast by Hostage Taker’s controller and the card will still be returned to its owner’s hand if it’s still in exiled when Hostage Taker leaves the battlefield.

Yes.

The biggest interaction is with cards that remove abilities after effects are applied to creatures. When a card gains a keyword, it actually gains an ability, whereas something like “can’t be blocked this turn” actually changes the game rules while the effect is active.

Yes.

Once Bruna’s ability is on the stack, it exists independent of it and removing the ability from the permanent that was its source doesn’t cause the ability on the stack to cease to exist.