Temporal Breach

What happens when I play Temporal Breach targeting my Elder Dragon in base and pay [A][A] for Dragon Roost while playing Elder Dragon? Will it be played to base or to Dragon Roost?

You can choose which of those two locations you want to play it to.

In general, there are not many cases where we have this kind of competing instruction: one effect that tells you to play a unit to one location, and one that tells you to play it to another. That being said, when they do arise with contradictory requirements, the precedent from other areas of the rules is that the player chooses one of those requirements and ignores all others.

Note that this only applies to mandatory positive requirements, ones that specify a location or method without also preventing the action from being performed to other locations or with other methods and that aren't optional. For example, an effect like Perched Grimwyrm's passive ability is a restriction on playing that unit to any location other than a battlefield you've conquered this turn. Rockfall Path's passive ability is a restriction on playing units to Rockfall Path. If you've conquered only Rockfall Path this turn, you can't play Perched Grimwyrm at all, because those two restrictions are contradictory.

In this case, the contradictory requirements don't directly prohibit the playing of these units to other locations, so we can choose one of the contradictory requirements and ignore the rest.

What happens when I play Hostile Takeover gaining control of my opponent's ready unit, move it to base, and then play Temporal Breach choosing it? Where will the unit go?

The unit will remain in Banishment.

Units cannot be played to a base of a player other than their own controller under any circumstances. Because the owner of the unit is the one being instructed to play the unit, and the location they are being instructed to play it to is a base other than their own, it isn't possible to play the unit. During the check legality step of finalization, the unit will be returned to banishment from the chain and all actions taken will be undone.