Can immigration documents (visas, PR, etc.) be picked up at a land crossing at any time? The website says borders are open 24/7 (obviously), but I'm not sure if there special hours of operation for picking up residencies/visas that have been issued.
No - visas cannot be picked up there. This is where you can go to activate a visa (e.g. land using a PR visa or activate a work permit). However you can't pick up the actual permit or visa from there.
here's a list of all border offices. You need to find one that does immigration, it has hours on it.
http://www.cbsa-asfc.gc.ca/do-rb/services/imm-eng.html