The Doctor said the cognitive test is the same one that anybody can find on line. I don't recall the Doctor using the word "Alzheimers." Someone in the press corp might have brought it up. It doesn't sound reasonable that the Doctor would plant that seed in anybody's head.
When the Doctor said "most people score a 26" I took that to mean the average score for someone of Trump's age is around 26. I doubt if the Doctor just pulled that number out of nowhere. It's probably a matter of record.
As for a psychological test, I recall the Doctor saying something about his close proximity to Trump's office allowed him to talk to Trump frequently and he didn't see any reason to doubt Trump's mental condition. He also said that if he ever noticed anything he'd report it immediately to the proper higher-ups.
I'm sure the Doctor told a couple little white lies about Trump's height and weight to prevent using the word "obese" in his report. I took a quick look at the body fat ration and Trump squeaked under by about 3 pounds.