You can search for a hashtag without using twitter by typing the hashtag on Google.

A hashtag is the pound sign that is followed by a word or phrase.

Send a tweet with the hashtag message included. i.e #whateverthetagishere

You can use the hashtag the way you normally do - just tweet it.

To know how many times a hashtag was used, you should track that hashtag using a hashtag tracking tool. As there are various tools, some of them which I know, such as :

1. Trackmyhashtag

2. Keyhole

3. Socialert

These tools provide free trials also, in which you can track your hashtag and get the insight that you need the most.

I hope you'll find your best tool.