These verbs are not synonyms. If you interrupt someone/something, you don't necessarily disturb the process that is going on. Like when you say:
I'm sorry to interrupt you, but I have a question. - You'd say this if your professor was giving a lecture and you needed some additional explanation, or wanted to make a remark. To interrupt means to pause a process/the stream of conversation, usually only temporarily.
I hope I don't disturb you. - You'd say this if you felt that your presence might be unwanted at the time. If something disturbs you, it really bothers you. You can say that shocking images of violence on TV are 'disturbing'.
To hamper means to make something more difficult, harder to do or achieve.
Loud noises may hamper your concentration.
She's talented for dance, but she's hampered by her shyness.