How long do typical battles last (as far as plot/PC battles go)?
In theory it should last as long as modern battles do because of attack waves, multiple line defences and attrition, but in practice as most Nep plots have their pilots as part of an elite special operations strike group, theirs is just to go in, fk some shit up, and get out ala SST Marauders, PA battles in those cases are quick, personal and bloody.
How is the physical toll as compared to, say, fighting in just clothing or naked?
Depends on the PA, some are entirely bare bones for pilot comfort (Demon types come to mind), some do not even fully substitute the pilot's own strength (Nep ID-SOL suits). But then you get to sit in a nice controlled atmosphere instead of being in smoke, dust and alien NBC agents outside in god knows what planet you happen to be attacking/defending at the time. IMHO the physical toll would be less (But the mental toll would be higher because you have to keep track of all the data being sent to you and stuff) Though you'd probably be shuffled around to all the hotspots on the battlefield by command until you expend all your consumables, become too damaged or die, just so that maximum use can be had from the suit.
Even if the PA does reduce physical strain below basic levels, do pilots get sore from combat due to long combat duration, or pull muscles due to the wide range of motions a PA might do for aerospace combat maneuvers?
You definitely get sore and chaffed because of rubbing on all the wierd projections inside the pilot compartment (Suits will never fit you 100%) That said, suits will also never protect you 100%, some kenetic energy attacks dispite not penetrating the armor will still have some force transmitted on you, which can range from pat to a rib-breaking punch. You'll also get tossed around a lot by explosions, melee combat and thruster use, mitigated by the effectiveness of any padding or damping system inside the suit.
EDIT: Forgot to answer a part of the question. I do not think it is more likely to pull a muscle in a PA compared to fighting unarmored. Firstly usually all of the weight is being borne by the suit, secondly your range of motions would be inherently less than unarmored due to the nature of wearing some kind of suit itself (You can't breakdance in Gothic Plate regardless of how strong you are or how light it is for example). Considering that muscle pulling is based on these two factors, you get the idea.
Are pilots pumped full of stimulants to temporarily override fatigue and sharpen combat senses? Or are they pumped full of depressants to prevent adrenal effects from exhausting the body?
Most suits have provision for automatic or manual application of drugs, including the E/M Series of Nep suits. The Nep military however have a policy of not discriminating between clean/un-cyberdized soldiers and druggees/full conversion cyborgs. Any equipment requirement and treatment are with regards to the least restrictive option unless it is to save a life. To summarize, use of performance enchancing drugs and cybernetics are up to the individual soldier, but there are cases in life threatening situations to save a life, for example where a biological organ for transplant is not available, the use of sedatives in medical surgery, or with regards to your specific question, a squadwide stimulant was automatically applied in Acadia Mission 2 when the entire squad was dazed from a massive explosion during a critical period.
EDIT: Misread Kai's statement, thought he meant Nep suits had no drug injectors.
Do pilots ever "crash" after combat due to exhaustion from adrenaline and/or stimulants and sleep a really long time afterwards, or do miracle wonder drugs prevent this?
There is no helping it, examples of post-piloting fatigue are strewn all over the Alliance and Acadia RP. Maybe there will be better, side effect free drugs in the future, but for the morality of the site and for game balancing mechanics, drugs will always have (probably undesirable) consequences. Also, note the nep non-discrimination policy with regards to drugs explained in the previous question.
Do pilots typically come out of PA all sweaty, or are these effects also prevented due to self-cleaning suits or something?
I think Yam suits are self cleaning, but I have not designed Nep suits to be such. Better scrub out that padding or whip out that Dr. Scholl because that cavity is going to smell like an old sock.