[RPG] Can clerics apply metamagic feats to domain slot spells

You would cast a 4th-level Ray of Enfeeblement, and it would cost a 9th-level spell slot.

Without Heighten Spell, it would be a 1st-level Ray of Enfeeblement costing a 6th-level spell slot.

Metamagic does not increase the effective level of a spell. Instead, for powering up the spell with other effects (like chain and empowerment), you pay the price of a higher-level spell slot for a lower-level spell. If you wind up with a 1st-level spell taking up a 6th-level spell slot thanks to metamagic, it's still a 1st-level spell.

Powering up the effective level of a spell is exclusively the role of Heighten Spell. Think about it: if Chain Spell meant you were also casting an effective-level-4 Ray of Enfeeblement, what would be the point of Heighten Spell even existing?

See the bolded text here, in which it increase the effective level of a spell - emphatically, unlike other metamagic feats:

Heighten Spell [Metamagic]

Benefit: A heightened spell has a higher spell level than normal (up to a maximum of 9th level). Unlike other metamagic feats, Heighten Spell actually increases the effective level of the spell that it modifies. All effects dependent on spell level (such as saving throw DCs and ability to penetrate a lesser globe of invulnerability) are calculated according to the heightened level. The heightened spell is as difficult to prepare and cast as a spell of its effective level.

RAW: No

In short - extracts are not spells. An alchemist does not have "spells", despite the many similarities extracts have to spells. The RAW answer is a no.

Note that the alchemist's extracts are also not potions. So although metamagic could be applied to potions, that isn't meaningful here.

From the Creative Team: No

If that answer doesn't satisfy you, consider that James Jacobs (Creative Director) answered this question in a "ask me anything" on the Paizo forums:

Danny Kesler: Can [an alchemist] select and benefit from metamagic feats? What about metamagic rods?

James Jacobs: No.

Source.