You should water 2 or 3 times a week if the weather is hot.

This sounds harsh but you really, Really, REALLY should have written in before doing anything. All that sod under the trees is wasted. It will might survive until winter, but it will not return at all next spring. Bermuda needs 5+ hours per day of DIRECT, UNFILTERED sunlight. Sometimes that isn't enough.

Tifgrand is an expensive hybrid bermuda with very nice characteristics. Seeded bermuda is a wild grass with no nice characteristics. Seeded bermuda is the grass that most bermuda haters hate. So you just blew it all - everything you invested in the Tifgrand is basically lost either to the shade or to the seeded grass.

You say you watered it daily at first. If, by that, you mean you watered lightly 3x per day, without every letting the soil become soggy, then you did that part right. If you watered once a day, heavy or light, then you're not going to get a great start.

You say you've used a high N fertilizer a couple of times. That was wasted, too. The farmer who grew the grass dosed it with high nitrogen fertilizer before selling it, so it should have had enough to go a full month.

Lawnmower should be fine on the sod as long as the underlying soil is not soggy. If your feet sink in when you walk on it, then let it dry out for several days before doing anything.

As for disease, bermuda almost never gets disease.

Cannot comment on over or under watering it because we don't know how long you are watering or how often. Generally bermuda sod can be considered to be fairly mature in a month, but yours looks pretty dead. I'd leave it alone, no water at all, for at least a week and see what it looks like. Bermuda is hard to kill, but if it gets too much water, that's what it looks like. In a week you can put down 1 inch of water all at one time. Measure 1 inch using cat food or tuna cans placed around the yard. Turn the sprinkler on and time how long it takes to fill all the cans. That is 1 inch and the time is your new watering time from now on. Then don't water again for a week. Again, it's really hard to kill bermuda. After a week it should be greening again. Unless you get rain, then water once a week until the temps cool into the 80s. Then back off to watering 1 full inch but only once every 2 weeks.