First off you are going to want to read information regarding best watering practices. You want to water less frequently... and more at a time. Right now you are encouraging roots to only be at the surface...

Like a houseplant - you want to water once the soil is mostly dry. Then give it a thorough soaking down to 4-6"... then let it dry again. Roots will be encouraged to go deep to find moisture and nutrients. "Deep and infrequent" is the name of the game. Measure water delivery and shoot for 1" per "watering" , once a week. This will take some time to get the grass trained/conditioned to it. When you see the grass show signs a stress (blue grey... foot prints remain for long time, etc) then you know you need to water.

... to the real question, mowing heights... depends on type of grass. When it is hot, longer grass helps keep soil temperature down, encourages longer roots, and lowers need for water. Typically spring and fall you can cut shorter. For tall fescue, northern mix, or kentucky bluegrass.... 3" is common height, maybe 3.5" if very hot. 2.5" for spring/fall...

Yes, you don't need to water that much! Grass is very hearty, btw. (also, I hope you're not in California!).

Agree also with BoatDrink re: 3" height in general for grass. In the growing seasons (spring/fall), use a bag to collect the clippings as you mow. In the hotter summer months, don't use the bag as the clippings will act as a mulch and help keep moisture from evaporating.

Assuming in zone 7a you have fescue. My lawn is 100% TTTF and my mowing heights vary throughout the year. In the fall (early September) I start bringing the mowing height down. By winter and the time my grass stops growing (usually mid to late December) I'm down to around 2 to 2 1/2 inches. Come spring I start out at 2 1/2" again and slowly come up to 3" by late April and then 4" by mid May depending on our daytime temps. Once temps are forecasted to be in the upper 80's I'll bring it up to 4 1/2" which is as high as my mower will go. Personally, I prefer the look of fescue when it's in the taller range and IMO being taller during higher temps helps out quite a bit. With our temps being in the 80's already I went to 4 1/2" this weekend and it will stay there until early September.

We can't really answer your question without knowing what kind of grass you seeded. If you seeded bermuda then you are early. If you seeded fescue, rye, or Kentucky bluegrass then you are so late it's crazy. So we need to know what you seeded and where you live. Please be specific about where you live. Nearest town.