Re-using oil pump and pick up

Just my opinion... Toss the pickup because you can't clean in thoroughly enough.
Rings: if you have a spreader and can take them out without them twisting at all, then I'd say reuse them. Also IMO, you cannot properly clean those parts without taking it all apart. If you do decide to replace the rings, I would not use a general use hone. Either have them properly re-honed or use some green scotchbrite over the square stones on a flex hone and very lightly de-glaze the bores. I've had great results replacing rings and doing that. Remeber the bores have to be ultra-smooth for moly rings. Not hand hone or stone can keep them that smooth. the scotchbrite just cleans the crap out of the existing hone marks, similar to a plateau honing brush.