Aeration improves circulation around roots and allows deeper water penetration into the removed cores. Ordinarily you do it around feeding time but it generally will not hurt the turf but for the appearance of it looking like 1,000 rabbits left pellets on top of your lawn. Of course if the soil has been dried out by drought the aerator will just bounce along the surface and never penetrate to pull out the cores as desired.
De-thatching is more aggressive as the tools need to beat up the grass blades more than a little to get at and yank out the thatch. I would not recommend it at this time unless you have more than adequate irrigation to compensate.
As for your brown spots? Make sure you mow your turf tall to help a bit with that. You can buy a hose end aerator that may help. It has two water jets on either side of a plate you put your foot on.
Brown spots, if all else is healthy and the appear for no particular reason could be insects (e.g. grubs) or a fungal growth. Water early in the day to help prevent the latter. Apply an insectiside to kill the grubs.
You know, depending on where you are, stop by and talk with the parks and rec or even golf course turf managers about such questions. Was one once and was always flattered people would stop by to ask questions now and then. If your state has any money left for things like ag extension? Call those folks also. California used to have a nice group dedicated just to turfgrasses and related consumer questions. I suspect gone now with the budget crisis but the publications might still exist. Turf has not been an issue for me in the decades I have lived, on and off, in Chicago. I did ask for a riding lawnmower to navigate the streets of Manhattan but nobody ever gave me one for birthday or Christmas.