The brown spot is probably from overwatering. Cut it back and the situation may resolve itself.

Light, frequent watering encourages very thin, shallow root systems that can't tolerate a drought. More or less, it doesn't grow roots because it doesn't have to.

During hot periods...most of the time where you are, I think!...twice a week is very much enough, 0.5" to 0.75" at a time. Cooler periods are around an inch all at once, once a week. Subtract any rainfall from that total.

Clay soils do have trouble absorbing water. Mine does. Fortunately, for a lawn that small you can use one single sprinkler and cycle it, 15 minutes on and 15 off, to give the soil time to absorb. On the up side, clay soil holds a LOT of water once it does get wet. The time you water doesn't matter so much, but with brown spot please avoid watering in the evenings. That just encourages it!

For now, stop watering. Once the grass shows signs of stress (when you walk, it doesn't spring back up in a few minutes), water it as above. You'll have to test to see what you're putting out--a tuna can works well, or an old coffee can.

You'll slowly adjust it to longer waterings, deepen the root systems, and waste a lot less water. Quite a bit of the 5-7 minute watering is evaporating off and getting lost.

For seeding, the rules differ--2 to 4 times a day watering, about 10-15 minutes at a time to get the seed going.

Fertilizer...a soil test would tell you what you really need, but barring that a good balanced fertilizer in the 4-1-2 proportion is a good bet. 16-4-8 would be perfect, but if you can only find, say, 24-2-6 then use that. Just get as close as you can.

Fescues are already some of the most shade-tolerant grasses, although others will have much better suggestions than I would. I'm a Kentucky bluegrass person myself.

With a yard that small I wouldn't bother with seeding....just buy new sod

So if I reseeding it will interfere with my schedule watering then. I plan to cut back and may be water about twice a week from now on and reduce to once a week when it get colder. I hope after fertilized plus adjust water schedule and cool temperature the grass will create new growth.

Thanks
James

BTW I plan to buy only one fertilizer for all my plants due to I don't have enough storage. Is that 16-4-8 suitable for fruit tree as well? I do have bananas, raspberries, tomatos, jujube, some herbs, and gardenia. I also have 5 roses but that I bought fertilizer made just for roses.

the grass will create new growth.

Fescue isn't known for spreading very much, and it doesn't tiller out like bluegrass does. Yes, it'll grow some, but probably not enough for you to be happy about it. Reseeding's the only way to do that.

Is that 16-4-8 suitable for fruit tree as well?

I wouldn't think so, but I am no expert on fruit trees. If memory serves, most flowering plants, at least, want a lot less nitrogen and a lot more phosphorous. My annuals and perennials get a 1-1-1 balance, the grass a 7-2-1.