About the alternator. Today's cars will NOT run without a battery. The battery is needed to excite the alternator's field coil. It may work for a time but any increase in load will cause the alternator to shut down.
No battery = no alternator
Besides, it just makes good diagnostic sense to look at the basics before digging deeper. I'm constantly amazed at how often ppl will ignore the "Corrosion Cauliflowers" going on the battery terminals.
What are the tests results?
2003 GT Convertible (sold & missed)
2000 GT coupe (Craigslist project. Fixed. Now my DD). Windsor to Romeo swap.