Usually "doze off" is used to mean "falling asleep when you don't intend to". But "drop off" would be used more in a situation like "I have to drop my son off at school before I go to work". So, you have the meaning right and I think "doze off" sounds more natural and makes more sense for the meaning you described. :D

Yes, they're the same. The phrasal verb 'to drop off' has several meanings, and this is one of them.

Perhaps in England they use 'drop off', however in America, we don't use that term. As for 'doze off', the meaning is exactly as you said!