yes, they're similar and all basically mean the same thing. A few slight differences though for a native speaker:
1. 'bother' lightest & most casual form, usually used for asking for something.
2. 'annoy' has negative connotations (as if you know the action might make them get angry)
3. 'disturb' same as 'interrupt' this usually implies they are doing something beforehand
4. 'trouble' another casual form, very similar to bother.