Cloudflare is a multi-featured reverse proxy. CDN is only one of the features, but you also get some other optimizations and security features.
If you only want CDN, there are many CDN-only providers out there, though those usually charge for subscriptions, but do hold onto you cached data longer and some even offer Push (active caching) functionality.
Though all the extra features of Cloudflare are pretty good.
To answer your questions:
See above. But you can tweak the cache settings. Do a search for Wordpress and you’ll find many posts on the subject.
The Cloudflare plugin offers different functionality than Wordpress caching plugins.
To use Cloudflare, you need to use Cloudflare’s Name Servers.
That error may indeed be caused by another plugin that’s linked to your Cloudflare account. W3TC may be the cause if you entered your Cloudflare credentials there.