Q: car shakes badly after start when sitting briefly but is okay after sitting hours

My son went to his apartment for lunch yesterday and after 45 mins he started it to drive back to work. It was shaking so violently that he could not drive it. He said, "feels like the engine is going to fall out". He got a Uber rider to work, then home, and started it again. It behaved normally (it had sat for hours). He drove it to AutoZone and left it running while they checked the codes, which indicated cylinder misfire and bad fuel injections (P0202 and P302). He bought a cleaning product and filled up with gas, but the car shook violently again at starting at the gas station. He waited an hour and it still shook, so got a friend to pick him up and went home. He went back this morning and car started normally and he drove it 30 minutes to allow the cleaning product to mix with fuel. He is waiting to see if this product helps, but I'm concerned.

My car has 115000 miles.
My car has an automatic transmission.