In Diablo II, there was a cap of 75 set on all resistances (though certain items could increase that cap). Is there a similar cap in Diablo III, or am I free to increase my resistances willy-nilly?
There is no cap on resistances, neither is there a penalty in higher difficulty levels; that's because the resistance value itself is no longer a simple percentage but instead behaves like your armor - it's used to determine a percentage-based damage reduction, but is in itself an ever-increasing number with no cap.
In addition to that source, you can open your character's details page, hover over resistances, and see that they define a percentage damage reduction but are not directly that percentage themselves. You can also see it changes depending on the enemy's level, just like the damage reduction from armor.