Will a GFCI-protected 15A circuit meet light power tool needs in a shed?

Your dedicated run of 12/2 could handle a maximum breaker size of 20 amps and you'd need GFCI protection in the shed. If the wire size for your existing GFCI is 14/2 then the maximum breaker size is 15 Amps even if you run 12/2 to the shed from that GFCI. A small power tool, lights and a stereo shouldn't be a problem if the existing circuit was lightly loaded and wouldn't be a problem with a dedicated one.

If you're running PVC anyway...there's no need to wrestle that UF alligator down the pipe!

Since you're planning to run PVC conduit in any case, you shouldn't bother trying to wrestle that 12/2 UF alligator down it. Instead, what you want to run inside the conduit are 3 individual 12AWG THHN/THWN wires (hot, neutral, ground) of the appropriate colors. This takes up far less room in the conduit, is far easier to pull down the conduit, and lets you use stranded wire, which also makes your life easier pulling things.

Running a fat conduit is a wise investment

Furthermore, if I were in your shoes, I'd make that conduit fat, so that the next person (who may be you!) doesn't have to dig it up and replace it if they need more power at the shed. A 1" conduit will easily accommodate a 60A feeder, so it'd be my minimum size for this; if possible, I'd go up to 1.5", which provides enough room for a 125A(!) feeder to the shed in the future, if such is desired.

Having GFCI upstream of outbuildings is never a good idea

However, there's no reason to provide GFCI upstream of an outbuilding, and a couple good reasons not to, primarily that you really do not want your lights knocked out from under you when something like a faulty power tool trips a GFCI. As a result, running from your panel with a conventional breaker at the panel and a GFCI receptacle in the shed is the best plan; it also works well with the fat conduit mentioned above. It does mean you'll need to make sure there's 18" of dirt between the top of the conduit and the finished grade once you've backfilled the trench, though; in order to do this, you'll need to trench down to 21 or 22".