First of all, the term CDP is not universal. Avis uses AWP, Budget uses BCP, etc.

Anyway, it depends on the promo and on the specific codes and also on the rental company. On some Avis promos (as explained in the wiki of the promo sticky in the Avis forum), you have to use a specific AWD to get the bonus. On other promos, you need to use a coupon code to get the promos. The number of airline miles you earn without any promo is pitifully small (not good for much except deferring expiration, which only matters if you don't regularly earn those miles any other way). So I recommend using a promo, and then following the T&Cs of the promo, which is often that you are required to use a coupon but the other code (AWD/BCD/CDP/etc) is just "suggested", not "required".

Still, you're safer off using only airline & hotel AWD/BCD/CDP/etc codes, because some codes from other companies specifically rule out earning airline miles (but if you're not familiar with the code, you might not know that!).

Btw, in case you want to earn UA miles with Hertz, most UA Hertz promos (that give thousands of miles per rental) require you to book through the UA site, not through the Hertz site. (But you can't enter your flight information at the UA site, so you have to come back to the Hertz site where it tells you it's missing your flight information and gives you a chance to it.