As noted, dogs are very restricted in the national parks.
Arches and Moab area is hot in July and August. Average high is 96 degrees. Temps above 100 not at all uncommon. Sightsee early in the morning and late in the evening. One hike you can do with a dog is Negro Bill Canyon about three miles up UT 128 from US 191. The trail is in a canyon so shaded from the sun part of the day. A stream flows through the canyon for the dog to splash. Keep the dog on a leash as they report poison ivy in the canyon. There is an interesting arch at the end of the trail. The LaSal Mountain Loop up Castle Valley to the LaSals is a good way to spend the middle of the day. Quite nice up there with great views overlooking the red rock country.
Grand Canyon is higher up and thus cooler in mid-summer. Again, significant restrictions on where dogs are allowed.
I would think three or four days for GCNP and a week for the Moab area would leave you time to see other, cooler places. littlemo noted the San Juans as one escape from the heat. Black Canyon of the Gunnison NP is another park higher up in elevation. Bryce Canyon NP is the highest in elevation of the southern Utah parks and also worth a few days.
