The P is for P90 pickups. All emulation work through the piezo bridge and is identical on the 59 and 59P. It is in fact identical across all JTVS. The 89S come with a few different high-gain-oriented "patches", but the circuits are the same.
Acoustic emulation, just like a real acoustic, will sound bad going through either amp-simulation or a real guitar (pre)amp. This signal is designed for a signal path similar to one coming from a vocal microphone. It is designed to go direct to the desk. If you use a variax or a real acoustic connected to a line6 or other amp-sim, create dedicated patches for use with the acoustic sounds. You may use effects like reverb, delay, chorus and man others with great results, but amp-simulation should be turned off. Some simulators can't turn the simulation off, but instead offer a "direct" model for this purpose.
A JTV may be powered in 3 different ways: From the internal battery, via the variax digital interface when hooked up to a line6 device (POD or amp), or with a powerfeed through a stereo cable connected to a powered XLR/line footswitch.