If I use a free host to put up a site, can I register a domain after that and have it go to that site *without having to change any of the sites' coding* i.e. references to urls in the backend scripts etc.?, and without preferrably, the hoster having to offer any kind of "redirection" feature.

I'm a little curious as to why when you sign up the company gives you a "registered domain" or "free subdomain" option - well obviously with a sub, some conf will have to be done to their server to know that x.frog.com goes to /var/www/users/x or something, but is that all?

Yes, you can have a website be directed to another website - or just have the DNS changed so that a site points to a different location, and if the scripting of the site is all dynamic (pretty much just not referencing IPs) you should be kosher. I have a website, and when I bought a second name, I just redirected it to the first site, but it could have been the main site just as easy.

Not sure about your second question, I always thought that sub-domains were usually pointed to secondary servers, different machines.