We are finally moving into our new home. Due to construction issues (lack of water and vehicle damage), most of the grass has died back and there are areas of just dirt. Also, we have dogs.
Is there any variety of grass that I can plant now that will grow in the winter?
What would be the best material to cover bare areas during the rainy season to mitigate mud?
I would seed with perennial ryegrass! Perennial ryegrass loves the fall conditions, from September to late October. Seed at about 11 lbs/1,000 sq ft. You should get germination in about 7 to 10 days this time of year.
Apply some fertilizer high in nitrogen on the day of seeding at 1 lb N/1,000 sq ft. A good sample ratio would be 25% nitrogen - 7% phosphorus - 15% potassium, or 33-3-12. High nitrogen, low phosphorus, and potassium in-between.
Cover the grass seed with mulch no deeper than 1/4". The type of mulch is relatively unimportant, although finer mulches often work better than coarser ones. Fresh sawdust, aged sawdust, fine fir bark mulch, compost and ground-up grass straw can work well.
I would be sure to reseed every 2 or 3 weeks until you have a nice dense stand, your dogs will disrupt establishment and a single seed application will not provide the results you want. If you could fence off areas while establishing (for instance half the lawn) that would help as well. Sod is the other option in your case.
For more info on turf renovation, see the publication Practical Lawn Establishment and Renovation.