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It is entirely possible. You will want to make sure you get hosting with unlimited subdomains. Some shared hosting providers limit the amount of subdomains (not sure why, it doesn't cost them anything). Anyway, you can then redirect traffic from the subdomain to another domain instead of locally hosted files.
You would basically do what is referred to as a 301 redirection. This will redirect all traffic from the source (your subdomain) to the destination (the remote website).
However, as far as I know, redirected subdomains will not appear in search engines because the original purpose of them was to redirect traffic to a new domain because the old one will be expiring or no longer being used.
So what about retaining my domain in the browser? Is there some scheme maybe with frames that I could do that with?
Frames are no longer advised as support for them will be dropped in the future (already deprecated in html5 spec) so that isn't a good idea. If you want to keep the original URL in the browser, you would want to use rewrite redirection. However, as soon as the user clicks a link on the remote site, it will change the url to the link they clicked on.
Ok that sounds good enough. Not what I hoped for though, but I can work around this I guess. Thanks.
You may be able to use a proxy to keep the original url but I have no personal experience with this.