If I submit a domain that I have redirected (i.e., forwarded the URL), will search engines index it based on the redirection site, or will they not index it at all?
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yes, depends on what kind of redirect are you talking about, 301 is fine, but don't use meta-tag refresh, since search engine will hate it a lot (think your website is a spamming web site) so use 301 with .htaccess or mod_redirect.
What I have read is that google handles the 302 also very efficiently.
"From a search engine perspective, 301 redirects are the only acceptable way to redirect URLs. In the case of moved pages, search engines will index only the new URL, but will transfer link popularity from the old URL to the new one so that search engine rankings are not affected. The same behavior occurs when additional domains are set to point to the main domain through a 301 redirect."