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A new battery may help, but I'm not sure. I say this because I store my car every year with the battery out of the car for 4-5 mos. and my fuel gauge is always accurate as soon as the car is restarted for the season.
so far as the pump cutting off prematurely, this is entirely possible. There is an issue with these cars with the hoses inside the fuel tank. It is covered in TSB Boxster 3/99 2015 2.
One of the symptoms of this issue is innacurate fuel readings. It was first discovered when owners began to run out of fuel while the gauge was indicating fuel in the tank, and when reporting some difficulty in fueling the car.
The fix is to remove the fuel tank hose assembly and, using nylon cable ties, to secure the hoses in a way that they do not interfere with either the inner flap
of the fuel filler neck or with the travel of the fuel sending unit arm and/or float.
PM me with your email, and I'll send you a copy of this TSB. It can be done DIY in under an hour...with all the usual cautions about working with the fuel tank opened, etc.
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