Yup

Both require attunement, but you've got three attunement "slots" so there's no problem there. One requires the wielder be a Warlock, the other just requires the wielder be a spellcaster. So no problems there.

Looking elsewhere, the only conflict we might run into would be "Multiple Items of the Same Kind" (DMG p.141). But that's directed at things—like layering two pair of gloves or boots—that would be physically incompatible. Holding two one-handed objects really doesn't seem to run afoul of it.

If this seems overpowered, just remember what it's "costing" the warlock: two of her three attuned items, along with whatever it took to get two very rare magic items, plus a free hand for material components. And it only benefits her spells which require attack rolls. It's a lot, but it's not worse than a rogue who got her hands on a +3 longbow and a Manual of Quickness of Action. The point is: if your game's got very rare magic in enough quantity for a single character to grab two harmonizing items, that's going to be a lot of power.

No

DMG p.139:

A scepter or just a heavy cylinder, a magic rod is typically made of metal, wood, or bone. It's about 2 or 3 feet long, 1 inch thick, and 2 to 5 pounds.

A quarterstaff is about the same weight but 6 to 9 feet long i.e. more than twice as big.

Essentially, a rod is a "stick" - you could try to make a case that it is similar to a club but clubs generally have weighted ends, albeit not as weighted as a mace.