How does a rod work when it comes to spells vs. spell slots? Let's say I have a Persistent lightning bolt memorized (+2 levels, in a 5th level slot). Can I use my lesser rod of empower on the 3rd level lightning bolt, or will it fail if I try to use it on the spell-in-a-level-5-slot?
You can use a single metamagic rod with any spell you've applied your own metamagic feats to. The metamagic rod does not change the spell slot value, but the metamagic effect applies.
Edit:
I'm not sure if you can use a lesser rod on a spell your metamagic feat has turned into a fifth level spell. I don't think the rules are clear. I would probably rule that you can't, but I bet others would rule otherwise.
Oops, never mind. My mistake. Metamagic feats other than "heighten" don't actually change the spell level, so, yes, you can use a lesser metamagic rod on a 3rd level spell that you have used a metamagic feat to need a fifth level slot to cast.
But you couldn't if you used the "heighten" metamagic feat to boost it to a higher level.
How does a rod work when it comes to spells vs. spell slots? Let's say I have a Persistent lightning bolt memorized (+2 levels, in a 5th level slot). Can I use my lesser rod of empower on the 3rd level lightning bolt, or will it fail if I try to use it on the spell-in-a-level-5-slot?
I would think that yes you can use a lesser meta magic rod in this situation. While the spell is using a 5th lv slot, it is still considered a 3rd lv spell. Meta magic doesn't change the lv of a spell (except for heighten spell), just the lv of the slot used. The only exception I could see is if you use the heighten spell meta magic to increase a spells lv (lightning bolt for example), then it would no longer be considered a 3rd lv spell, so you would need a meta magic rod that would affect 4th lv (or higher) spells.