I hear Overwatch multiplies your aim by 0.7 for the duration of the shot.

It doesn't technically modify your aim. Rather, the multiplier applies to the final shot percentage. This is to the attackers benefit as defense is proportionally reduced as well.

How does aim from situational perks, such as Cool Under Pressure apply? Does it get penalized as well, or applied straight on?

This one I don't know for sure, but my guess would be the penalty applies.

What about enemy defense? Does it get subtracted completely, or does it get multiplied by the penalty?

The reaction penalty is the last thing applied, so defense will have already been factored in to the chance to hit

When you dash, I hear overwatch aim gets multiplied by another 0.7. Is this true?

The overwatch penalty against a dashing target is 0.6. This replaces the basic 0.7 penalty.

What you describe is how overwatch penalties were intended to work in XCOM1, although a bug meant dashing provided no additional protection.

Important note: Concealed Overwatch ignores the aim penalty entirely.