How would you say this to a doctor?

Yeah, you could say that, or you could say "my urine was brown this morning."

Yes, you can use any color to describe your urine, if it is that color. If your urine was purple for some reason, you could say that your urine was/is purple.

Yes, they will probably think brown may indicate blood from an ulcer in the bladder/urinary tract or extreme dehydration.

Brown Urine is blood in the urine. You should go to the Emergency Room (ER) (a part of the hospital that handles life threatening emergencies) and see the Triage Nurse (a person with medical training that prioritizes medical cases, so that the people in the ER are treated according to how urgently they need medical treatment) who will evaluate your condition. Then, you will see a Trama Nurse, a nurse that specializes in traumatic injuries, a Physician's Assistant (PA), who handles administering medical attention until a doctor can get there, and the ER doctor, who is a medical doctor who specializes in emergency treatment. You would also likely see a Urologist, who is a doctor that specializes in the bladder and urinary tract, along with physical injuries to the kidneys. You might also need a Nephrologist, who is a doctor that deals with illnesses of the kidneys. Lots of exciting new vocabulary!

It would be "I had brown urine this morning". It could be "I have had brown urine since this morning", if you have urinated again and it is still brown.

You can say it was brown, or dark, or just "discolored." All are correct.