I love the PDC area, been there a few times. I would highly recommend Abyss Dive Center (
Playa del Carmen Diving with The Abyss Dive Center). They are just off 5th ave (1st ave between 10 and 12). The guide and boat crew are excellent and very knowledgeable and safe. One of the biggest reasons I recommend them is the personal service I have received every time. They always take small groups, max 6 people, but often I have dove with just my wife, the guide, and I. Abyss is owned & operated by a Canadian (from Revelstoke, BC if my memory serves me correct), and he runs as safe a diveop as any I have used in Canada.
There are lots of good spots on the reef that should be fine for both you and your ow certified buddy. Barracuda and Sabalos are great and max depth is about 50'. Tortuga is also a great dive and can be done shallow, or you can go deeper (100'), I've done both. Some of the best stuff I have seen on some of the shallower dives. When I was there in November, we had a dolphin swim up just as we were about to head up to our safety stop @ moc che shallow. Abyss can also help you with the centote's. Did those on my first trip, but my wife prefers the ocean with all the life. One suggestion I have regardless of who you choose to dive with is that, particularly if you want to do the bull shark dive, make sure the shop you choose doesn't chum or do feeding dives. Some of the shops do however. Abyss certainly does not.
Full 3mm should be fine for the wetsuit for the ocean. Cenotes are a little cooler water though.
As far as the afternoon dives go, I think most of the shops do 2 tank dives in the am and 1 tank in the afternoon, although if you request it, you can probably get accommodated for 2 in the pm. If your looking for someplace for lunch, check out wicky's on the beach. Its just around the corner from abyss, and was excellent (didn't find it till my last trip there, and man was I missing out).