1. Is there a way to get Internet access in Safe Mode?
No. The modem drivers can't load. No hardware drivers can load in Safe Mode.
2. Is there a way to stop the computer from shutting down long enough for me to update and scan (which may be several hours)?
Well done on your progress to date. If it is an overheating problem: -
(a) Open the case to improve air circulation. Don't touch anything inside!
(b) Alternatively, observing proper anti-static precautions, buy and fit a case fan to improve cooling.
3. With the drive connected to my good PC, what can I do to clean or fix it other than virus scan, trojan scan, etc.?
Many things. See this page:
http://myweb.tiscali.co.uk/bridip/recovery.htm 4. If I re-install Windows 98, how will it affect the registry? I.e., will it eliminate bad registry entries?
It will eliminate all your software, and all your hardware too. Windows will no longer be able to find the device drivers needed to run any of the hardware peripherals, e.g. modem, printer, scanner, because you will have deleted them from the hard disk. And all the installed programs will not work, except Windows itself. It is a suicide option.
Slightly less rash is to obtain an upgrade version of Windows 98 on CD, i.e. one that only installs a new C:\WINDOWS directory, rather than a full install version that wipes your whole disk. Then rename C:\WINDOWS folder to C:\WINOLD in DOS first, before doing the install. That way, the old driver files get preserved, not deleted. And you get a new C:\WINDOWS directory, but if necessary you can delete it and put the old one back (by renaming C:\WINOLD back to C:\WINDOWS). And when the driver installation wizard asks for files you can point it to C:\WINOLD and its subdirectories.
5. Any other advice on how to work on a PC that keeps shutting down?
You can't! The only viable option is to disinfect the virus, and any other malware present, and then to make full use of the advice at point 3 above.