How to test whether the proxy is working?

It might be a noob question but how does the one test the proxy that has username and password whether it is working or not?

Also, how to test it against a specific sites (like Instagram or Twitter)?

To actually use them. Just use foxyproxy (firefox) or proxymega (chrome) to test those proxies with userw auth.

You can either use a script (via cURL) or through your browser by manually inputting every proxy into a proxy manager extension (like foxyproxy).

But, if you think of testing your proxies prior to buying them, then there isn't much you can do. You can't really test them before they are allocated to you.