To answer the main question:
No, the attack of a chaotic character does not count as chaotic.
A chaotic character who wields a weapon does not make this weapon chaotic. From the point of view of the universe, a second level character does not change the nature of the weapon he handles.
A Anarchic The weapon is impregnated with the power of chaos. Makes the
chaotically aligned weapon and therefore prevents corresponding damage
Anarchic is a +2 enchantment. Without it, or an appropriate skill, the weapon that is handled remains like an inert piece of metal / wood / whatever (and therefore does not inflict damage aligned to chaos, therefore, will not stop an inevitable regeneration).
Ways for a second level group to gain chaotic damage:
- Anarchic Stroke (minor) of a first level Warpriest
(No other spell analyzed with descriptor (chaos) has yet been found)