Depends on what the job is, are you going to be the Dean, the janitor, something in the middle?
I'd be pushing for full family status, i.e. medical coverage for your wife included, but seems strange they would offer school fees but not that. Should be fairly easy to negotiate if you say you're happy to drop the schooling benefit - you can renegotiate that after 6 years if need be. See if they'll throw in a relocation allowance too to soak up some of the cost of shipping the pooch, they can only say no and I'm sure they're not going to be offended if you haggle.
For your budget and being dog friendly, you're probably looking at a small villa somewhere like The Springs or Mirdif (to name 2 of many possibilities) but it would really depend on the location of the university.
You'll probably want to push your wife to work too. You'll get by just fine on 29.5k/month but if she can bring in an extra 10-15k/month (she should get that if she's British and has some experience doing any kind of office work) you'll be living very comfortably. Plus it'll give her a sense of purpose and all that jazz.