About 7 large oak trees in yard. Applied at the wrong ratios will damage your trees. The more mature they are the longer it will take to see the damage, maybe even years later. I personally do not like treating the whole lawn for weeds except with preemergents. When it comes to post emergent applications I only spot treat.
Sod is already grass with roots.
It will root faster for the dogs but will take 2-3 weeks. Hydromulching, in certain areas at certain times of year will be ready in three weeks. This is not always true. Depending on the type of grass, soil temp. I would need to know what type of grass and where you are to give you a more detailed answer.