When you say "DC motors", I think you are meaning to say "commutated" ie the field in the rotor is rotating relative to the rotor itself – and the field on the stator is stationary, so frequently the stator uses permanent magnets.
By the same definition when you say "AC motors", you are probably referring to machines where the field rotates within the stator and the field within the rotor is static (or almost so in the case of the induction, asynchronous machine). This includes wound field machines such as clawpole alternators.
The universal motor is essentially a "DC motor" in this unconventional definition, just to cause confusion. Possibly one reason why normal convention doesn't follow these definitions.
Essentially these 2 basic motor classes simply refer to the same electromagnetic concept but one is the other turned "inside out". The conjecture about maximum speeds due to lamination sizes applying to one but not the other is simply wrong I'm afraid. High speed brushed motors require the rotor laminations to tolerate rapid flux swings without silly losses just as much as the stators of induction or PM motors, whether driven sinusoidally or trapezoidally (BLDC).
The rotor in the induction motor sees significant currents and flux is driven by current, not voltage. The fact that the rotor winding is shorted out doesn't alter this. However, the rotor resistance is generally relatively low, as is the consequent winding voltage.
The original proposition that AC is somehow better than DC is debatable. Depends on the context. As I may have mentioned above, a rotor with a permanent magnet will generally be more compact than one incorporating a winding which gives them a size advantage over induction machines. However, they are more expensive and complex to design and manufacture and will generate a voltage without external excitation. You have to consider what happens if the speed is enough that the back EMF exceeds the supply (battery?) voltage.
Interestingly, many hand power tools are now offering brushless versions for a slight premium. No brushes to replace, no sparks and no commutator to wear out. And more to the point, no significant change in motor size and power.