Is an animal shelter required to weigh the animal before being PTS?

You don't NEED to weigh, You can usually guesstimate based on looking at the dogs. There is a dose, but most people give more than that to be safe.
Edit: There is no law requiring weighing.
If you under-dose, they may fall asleep but not pass, so you simply give more.

Below is the euthanasia recommendations set forth by the AVMA. Yes it is best to weigh the animal before euthanasia. In the long run weighing is humane and saves money. This way over dosages and under dosages are prevented. I would guess that this would depend upon the shelter's protocols. however since you are in California there is most likely a law for this. So you will need to find anything that maybe specific for California. Below is a link to the California Practice Act. You can also contact the state veterinarian with your question.

As far as shelters euthanizing by request, I believe that depends upon the shelter's protocol and procedures.

that would depend on the shelters individual rules or policies

some do not euth.. some use heartsticks or gas.

i don't know the rules but do no that many times in shelter dogs are given too little and don't die..