You have multiple questions here, so I'll answer them one by one.
The search/tweets method
You can search tweets using this method in the 1.1 API. According to those docs, you can pass a q
parameter which is "a UTF-8 encoded query of 1000 characters maximum".
The query for your specific requirements
Scroll down on this page and you'll see that you can search for a specific hashtag using #hashtag
. Yes, that's it.
How can I make V1.1 requests using OAuth?
Follow the detailed step-by-step post here, using this library to make authenticated requests.