"I have finished (since)" --> Can I use since?
 

I have finished my homework yesterday so today I feel very happy---> Is this sentence correct? Some teacher have told me that I shouldn't use "I have finish" with "since". But I think that in this case, finishing homework is related to the present. And I want to mark the time in the past with "since".

Re: "I have finished (since)" --> Can I use since?
 

No - it's the verb form which is incorrect. Yesterday is a past time expression - and has no connection with the present. It's completely past and finished, and so you must use a past verb : I finished my homework yesterday and so I feel very happy.

If you want to emphasise that the past event (finishing your homework) has a present result (ie you feel happy), then leave out "yesterday" and use have + past participle (the present perfect) - the verb which expresses a connection between the past and the present :
I have finished my homework and so I feel very happy.

"Since" means "from a specified time in the past until now" and is used to indicate the time frame of an event which started in the past and continues until the present. Again, you have the past/present connection and so need the present perfect :
I have lived in Italy since 1983 ( = from 1983 until now)
I have had a headache since Sunday (= from Sunday until now)