Continuity mistake: When Raymond brings the doctor his milk and pill, the tray with the milk and pill are already on the bedside table.
Factual error: Columbo gives another detective $5 to go to the gun range for him. Technically, that is a bribe and it would cost both Columbo and the other detective their badges.
Factual error: All LAPD officers must re-qualify with their weapons multiple times a year. If indeed Columbo could not "hit the target," as he says to another detective, he would never have made it through the police academy, much less have become a detective and have been promoted to sergeant and eventually to lieutenant.
Answer: In the show, he was already a Lieutenant. A promotion would put him in the next rank up, which for the L.A.P.D. would be Captain. However, some of a Captain's duties would be overseeing other officers and ensuring they're compliant with policies, regulations, and standards. It would also most likely take him out of the field. This is something Columbo has no desire for as he rarely goes to police HQ's. Nor does he show interest in compliance and standards (for example, not going to his semi-annual evaluation at the firing range). However, he could still be assigned to a higher pay grade based on expertise, which is a form of promotion that does not include rank advancement. This would be going from Lieutenant I to Lieutenant II. I don't believe in the show it's ever started what his pay grade is. Although, in s02e01 (I believe) he mentions making $11K a year. Whether or not this was a true statement on his part, if you could find pay scale information for an LAPD Lieutenant in the 70's, it could give you an idea of his pay grade.
Bishop73