Certainly sounds like a battery failure - all sorts of weird things happen to the Porsche electrical systems when the voltage drops below 12 v.
When you change the battery, make sure you have your radio code handy or install a bridging battery so you don't lose your engine throttle adaptations and window settings.
I can't help you with battery pricing as I'm in Australia, but the Boxster has a rather large and good quality battery fitted as standard (Bosch / Moll), so someone more local can chime in here.....
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2001 Boxster S (triple black). Sleeping easier with LN Engineering/Flat 6 IMS upgrade, low temp thermostat & underspeed pulley.
2001 MV Agusta F4.
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