I rarely use SMS/MMS, so I didn't get a chance to try them directly...but I couldn't get it to work, based on the same loss of cellular data I first saw in 7.1.
Simply put, getting the mobile web on my MVNO didn't require using any config file (like from the carrier, or the unlockit site), prior to 7.1. The phone would just use the main APN setting entered during the SIM swap ("att.mvno"; it's the last one entered, and has always quickly reverted to "phone" after entered, but it still worked)
From 7.1 onward, including 7.1.1, if I don't have a config file loaded, the mobile web doesn't work, suggesting that none of the settings entered during the swap are being saved, presumably including the MMS settings.
There's always the possibility that I'm messing up the swap, but I believe the AT&T carrier 16.0 may be messing things up. IIRC 15.5, the prior version worked with everything, and was the version MVNOs stayed with even as the postpaid and direct users moved to 15.6 and later.
I'd love to be proven wrong...