“To be fair, let me say that Cloudflare made this site load faster. But it is not true all the time. And their “Always Online” feature is but a gimmick (maybe if you singed up for a paid account). How I wish it was for real!”
This feature is actually offered to all accounts, paid or not. The current product iteration does have some limitations that we’re working on with a complete overhaul of how this product works.

“. I checked the other sites not under Cloudflare but running on the same sever and they’re working fine. I thought it was always online. ”
This sounds like something at the host or server side was blocking requests from our IPs.

“I thought I’d be redirected to my preferred address format for that site but instead a paged that asked me to enter the captcha because my PC was possibly malware-infected welcomed me or my IP was some sort of blahblah. What’s weirder is that it kept on asking me to enter the captcha after correctly answering the previous one. Who wants that? ”
The challenge page is actually based on IP address only for a trigger (so…no difference between domain.com and http://www.domain.com). Why the challenge page appears is here: http://www.cloudflare.com/wiki/DataSources

Note: You can override this behavior by whitelisting the IP, which also helps correct false positives.

“Aside from those mentioned above, I also intermittently get 404 errors for pages that exist. I wasn’t sure at first whether this error had something to do with my host or with Cloudflare. I deactivated this site in the Dashboard and the 404 page turned into the actual page after a few minutes. ”
We generally wouldn’t do anything to cause a 404 error. Did you make any site changes, however? You could potentially have a 404 error if you changed some resource(s) on your site & didn’t purge the cache/go to Development Mode first.

“I added additional tag to these javascripts to exempt them from asynchronous loading but doing so moved my widgets to the footer area. Not to mention that my WordPress visual editor wouldn’t load. My solution was to turn this feature off.”
These beta features can cause issues & we recommend turning them off when you see a problem. We’re also working on building the ability for certain urls to be excluded from our proxy/cache, which is where we generally see the most issues from related to the beta features (admin sections of sites).