I think all of your expressions are used quite commonly. The first one 'Eating and drinking is not allowed on the bus' is not grammatically correct. I think it has evolved from an older phrase we used to see on buses 'The consumption of food and drink is not permitted on this bus' - where consumption is a single verb and 'food' and 'drink' are nouns, hence 'is' can be used - the phrase in effect saying 'it is not permitted'. Over time, the phrase 'consumption of food and drink' has been dumbed down to 'eating and drinking' The second sentence 'Eating and drinking are not allowed on the bus' is grammatically correct. Your third sentence 'Eating or drinking is not allowed on the bus' again seems incorrect. 'Or' is an expression of choice, so you can think of it as 'either .....or' - one or the other. So, I read the sentence as 'one of the actions (either 'eating' or 'drinking') is not allowed, but the other is perfectly acceptable'. In everyday speech, however, people do not use perfect grammar, so you will hear all these expressions.

Number 2 is the best answer, because there are two subjects, so the verb "to be" should be plural (are). BUT, you will hear native speakers make this mistake, so number 1 and number 3 almost sound like native English. If you are trying to write something for a public sign, go with number 2.

All of these are correct. Eating and drinking can go together as one and it still makes sense, but this may not work if you use other verbs.