Once the package is installed, the various files will go into their appropriate locations on the filesystem according to Fedora's guidelines for filesystem layout. Once you've installed a downloaded package, you can remove the RPM file.
You can see where a package is going to put stuff before installing it from the command line. Let's assume your downloaded package is foo-1.0-1.x86_64.rpm, and it's in ~/Downloads. You can see what's in it by running:
$ rpm -qlp ~/Downloads/foo-1.0-1.x86_64.rpm Where qlp is "query: list files from a packagefile". Look at man rpm for many other query options.
You can also use the less pager to see details about a package (it runs an RPM query command in the background), like this:
$ less ~/Downloads/foo-1.0-1.x86_64.rpm First, you'll see some overview information, then a description, then a (possibly very long) changelog, and finally at the bottom, the file list. (Hit q when you are done with less.)