I think i finished the Town of Mages quest before accepting Antimagic.
While i was in the 10 levels to finish antimagic i did the quest which needed rod of recall to finish. Using the rod didn't prevent me from finishing Antimagic.
You're allowed to wear anything your stats will permit. The limitation is on activating things, like wands. You're allowed to activate the Orb of Scrying for sure. I don't know whether you're allowed to activate the Summerlight Phial or not.
As far as I can tell you seem to be able to activate any artifact you have with no penalty. Unless there is something you cannot use but still doesn't deactivate Antimagic Shield and the other antimagic skills. Caveat: I've not gotten the Burninator with antimagic activated to test it, but both Rods are OK (I was kind of surprised, actually), as are the Vargh[sp] Redemption, the Summertide Phial, Eden's Guile, the Whatever-the-new-name-is-for-the-Arkenstone and (using) the Blood of Life.