It's not exactly the same, otherwise they'd probably just... use the same name.
They're very similar though. RedHat has more support and corporate-friendly options, generally, but those also come with a cost.
CentOS can be great for a server. Lots of big web companies use it to run high-availability environments.
Yeah, CentOS is free. You can download it and install as soon as you're ready.
I'm not sure what you mean by connect remotely.