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 with competing instructions: one effect that tells you to play a unit to one location, and one that tells you to play it to another. When they 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.