How do I use the sword rack? And can I put weapons on the racks on the wall?

Yes

How?

The Breezehome weapon mounts and racks don't seem to work for me. Though they do in my other home in Markarth.

you need to equip the item you wish to place on the rack then look at the rack in the location you wish to place the item, and hit your "activate" key.

For example for the wall racks you could equip the shield then point to the middle location(I think it comes up as saying shield placement or something similar).

For the standing weapon racks I think only some of them are usable.