I've done downloads from that server dozens of times on Windows 7 Professional. I've never tried with Windows 7 Home, but I'd be surprised if it didn't work.
RA does use a "download manager" utility called GetPlus, so your security settings might need to be tweaked a little bit. Watch carefully in Internet Explorer for a status bar or indication that suggests you need to allow that to run.
In the worst case, go introduce yourself to the local Allen-Bradley dealer and ask for their help burning a DVD with the free edition of RSLogix 500.