Are clerics allowed to apply metamagic feats to domain spells? Say I want to empower a 3rd level spell and put it into my 5th level slot, is that allowed?
The actual rule for domain spells says -
A cleric gains one domain spell slot for each level of cleric spell she can cast, from 1st on up. Each day, a cleric can prepare one of the spells from her two domains in that slot. If a domain spell is not on the cleric spell list, a cleric can prepare it only in her domain spell slot. Domain spells cannot be used to cast spells spontaneously.
Nothing there specifies one way or the other about metamagic. Applying metamagic to a domain spell, to make it use a higher level slot, does not in any way I can see break these rules.
Yes, but only if you use the domain slot of the spell's adjusted level.
If a cleric has at least one domain, then they gain one additional spell slot per level. This slot is separate from their normal spell slot progression, and can only be used to cast domain spells.
From the Domain class feature description:
A cleric gains one domain spell slot for each level of cleric spell she can cast, from 1st on up. Each day, a cleric can prepare one of the spells from her two domains in that slot.
So if a cleric wants to prepare an empowered version of their 3rd level domain spell, then they would need to use their 5th level domain spell's slot.
If they have the Quicken Spell metamagic feat (which increases a spell's slot by +4 levels), then they could prepare a single Quickened True Strike in their 5th level domain spell slot. However, this means they cannot use that 5th level slot to prepare other domain spells.