Dragon Roost

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 Perched Grimwyrm while I haven't conquered Dragon Roost this turn? Can I pay [A][A] and play it to Dragon Roost? What if I've conquered another battlefield this turn?

You cannot.

In Riftbound, "can't" beats "can." Perched Grimwyrm's effect is a "can't," as in it can't be played to any location but to a battlefield you've conquered this turn. Dragon Roost's ability tries to give the Grimwyrm permission to be played to the Roost, but that can't overcome Perched Grimwyrm's own restriction. If you have no ability to play Perched Grimwyrm to Dragon Roost after paying for it, you will not be able to finalize Perched Grimwyrm at all. The actions taken as part of playing Perched Grimwyrm will be undone and it will be returned to the zone it was played from.

This is true even if you've conquered another battlefield this turn. If you have the ability, you can play Perched Grimwyrm to that battlefield without paying the additional cost of Dragon Roost, but you can't play it to Dragon Roost itself unless you've conquered it.

Sharp eyed readers may remember that Baron Nashor doesn't work this way. He enters at the Baron Pit regardless of effects preventing him from being played to battlefields. The reason for this is that Baron Nashor's passive ability generates a delayed replacement effect, the one that reads "I enter there." This replacement effect replaces the event of him entering to any other location with him entering to the Baron Pit. The replacement effect gets around effects like Mageseeker Warden which limit where you can play units. However, Dragon Roost is not written as a replacement effect. It tells you where to play the dragon you've paid the additional cost for, but it doesn't specify how that dragon enters the board. Therefore it is a permission on play location, but not a replacement effect.

Relevant Rules: 054.2., 369.3.