If I've understood penalties correctly, very light, and possibly light, skills are unaffected by encumbrance, and it seems fatigue doesn't affect those skills either. If that's the case physician should be unaffected by those, and injury possibly likewise be ignored (quite important if you're close to death after failing to resist a robbery!).
Low endurance probably has a very nasty effect beyond slow recovery: namely an increased risk of fainting due to pain (my character before the current one died because of that: a wolf bite to the arm resulted in fainting... Well before that, most every robber encounter resulted in fainting as a moronic hireling would fire an arrow into the back of the character, and once a robber managed to hit with an arrow, resulting in the character being immediately robbed and transported away, despite the hirelings being in the way of them of actually reaching the character. They were more or less frozen in place as the character got replacement equipment and returned to the fight, zooming in to the same location as before the fainting).
Swimming distance is extremely dependent on encumbrance, while injury only has a minor effect. I haven't compared across characters, so I can't say anything about stat effects, although I'd expect a low endurance would cause fatigue to be accumulated faster.