Check around at your local nurseries. If they're selling flagstone, ask them if you should use concrete and/or grout for your flagstone installation. I don't see a problem as long as you properly handle water run-off. It can't sit on the patio nor along its perimeter. Be sure to keep this in mind when you do your walkway.
I don't see a problem using flagstone. Do this. Go to My Favorite WEB Search Engine, enter in +"flagstone" AND (patio OR walkway) and press Search. When you're done with looking at all the returned links, do the search over again EXCEPT put a ' ' (space) between flag and stone. In doing this 2nd search, you get another set of 'results'. Peruse them too. You'll find TONS of info on your project. I've included a few links for you to get started but DON'T just look at these links. Different folks categorize flagstone different ways. Some refer to them as 'pavers', others don't. Soooo, do the TWO searches to educate yourself.