As far as what kind of sod that depends on what is prevalent in your area. Without knowing where you live you are best to ask advice from your local full service nursery. For example, if you reside in northern latitudes, known to use a "cool season grass" a good sod that is available is a combo of Kentucky Blue/perennial ryegrass and fescue. The exact combination is up to the dealer but such combo, if you decide instead to seed, should provide an excellent lawn able to take what nature throws at.
If you live in a transitional zone...i.e. across the middle of America..Tennessee, Virginia, Kentucky...you get the idea....you can try using either a northern type..a cool season type or a more southern warm season type.
If you live in the south, then definitely consult your nurseryman and your neighbors who show good results with their lawns.
Southern grasses are completely different than cool season types. Planting either where they don't belong is a good way to waste money and effort.

Sod will not grow on top of old grass. That's the first thing you have to get rid of. Grass has to put down roots and it cant if it has to drill itself through hard clay soil holding old sod. Also, the clay soil as described, sounds like you have cement where grass should be.
Tilling is the only thing you can do but further to that you have to ...you must....incorporate organic matter to make the grass want to grow and hold on to moisture. Organic matter comes in many forms; i.e. peat moss, newspaper and compost. Without going too deep into this you should do research on it.
Without purchasing expensive compost, consult your soil dealer about what is called triple mix...a combo of different soils including sand.
Whenever you see the word "SAND" used in conjunction with gardening, always think 'coarse sand', not beach sand, or playbox sand...that holds too much moisture. Coarse sand is larger and allows for good drainage. Coarse sand can be bought at soil dealers or from quarries--the place where gravel is bought.
Drainage is one of the most used terms for good turf growth. Water should run away from the house.
So it sounds like you have to till and one of the easiest ways to do that is to rent a roto-tiller..the size is dependent on what kind of soil it has to bite through.
Into the soil which you dig down at least 6"...or more...you inject the organic matter. If you dig down 6", put at least 2" of organic matter in and turn it over. Then look to levelling and seeing to good drainage.
Weeds, if you see them, pick out and garbage them.
If your lawn is mostly weeds right now, I suggest you kill everything green there by the use of a glyphosate..i.e. RoundUp herbicide...it will kill the weeds, the grass presently there, anything and everything green so don't spray it around plants you don't intend to get rid of.
Wait the time mentioned on the label before tackling the job of lawn replenishment. One thing about the use of RoundUP, use it according to directions and follow it exactly. Here, more is a waste just like using not enough.
Moisture: Lawns require moisture at the right time. Usually that is 1" per week, rain notwithstanding.
Fertilizer: Set up a fertilizing schedule with a high nitrogen type that will help grass grow and greens it up.
Read the directions on the bag.
To understand how much to spread, simply divide the first number (nitrogen) into 100--that is the amount per lb per 1000 square feet, to use. So if the bag contained 20/6/12....i.e. 20 percent nitrogen, 6 percent phosphurus, and 12 percent potash, then 20 into 100 = 5 lbs per 1000 square feet. If a bag contained 40 lbs, then it would cover 40 divided by 5 = 8000 square feet.
Your small lawn...200 sq ft could be properly fed with 1 lb of fertilizer. But read the directions on the bag.

If you live in a southern zone, then throw out the info on using a herbicide until you find out what it will do on your grass type and soil. Speak to your nurseryman before you do anything. Some types of southern grass cannot be given weed killers--it will kill the grass as well.

Sorry about that missing where you said you lived...I swear Im blind sometime.
In California you definitely want to grow a southern grass.
Before attempting to give any worthwhile info you should check with your local full service nurseries about what kind of southern grasses do well in your particular area.
There are many types....and some, from reading in this journal, are much better....or much worse, than others.
So be sure to understand the difficulties of each. That wya, you can only blame yourself if you choose the wrong one.

The first mistake you made though was in not watering--even what grass you did have needed moisture. The ground needed moisture. But as you describe, the ground needs to be tilled to improve drainage and its ability to grow grass

A soil test would not be a thing to avoid. There would be probably many soil labs that do soil tests but the one you can check on is at your local--the one nearest you--state funded university extension service. You can find their location in the phone directory under "STATE" government, list of colleges/universities....dept of agriculture...externsion services. Many large nurseries also do soil tests.
A soil test, if you do some research, can tell you what your ground needs in the way of chemical imbalance correcting. It can save you many dollars in experimenting with all what is usually given to a lawn to make it grow its best.
Even a 200 square foot lawn deserves being treated right.

thanks for your input, goren. I guess I will have to till. Although, I'm afraid I might destroy the sprinklers in the process. Any advice on how to avoid that?