What are your high temps in the summer? If the temps get into the 90s, then seeding now is a waste of time. If they stay in the 80s, you might be okay with enough water. Most new grass roots are too tender to take summer heat until they are at least 6 months old. The sod should be fine.

If the houses are close together, the only grass that will work is a fescue. They are the most shade tolerant of the cool season grasses. Rye and Kentucky bluegrass need 6 hours of direct sunlight to really become dense. If you have a large unshaded area in the back yard, then a mix of rye, fescue, and KBG would give you the benefits of all of them. A triple mix like that is a little tricky to get started, but it is done all the time. All you have to remember is to not stop watering as soon as the grass comes in. You need to water every day for 3 full weeks to get all the KBG to germinate. The very best time to seed is late summer. For you that might be mid August after the warmer summer heat breaks.

With seed you get what you pay for. Ideally you will find a seed with 0.00 Weed and 0.00 Other Crop listed in the Guaranteed Analysis. The problem with weed and other crop is those seeds are the size of dust, so even a value of 0.23% weed could give you thousands of weed seeds.