1. The DAC only works on sound coming
from the phone, not for sound going
into the phone, so it makes no difference to the person you're talking to. (And a decent equalizer, like
Precise Volume (+ EQ/Booster) can do wonders for even a phone without decent receive audio [I'm extremely hard of hearing, and it makes phone calls a pleasure again.].)
2. How strong the carrier's signal is (or the Wifi signal is on T-Mobile Wifi calls or Hangout Dialer calls) is a lot more important to call quality than the phone. Although Verizon covers more
people than the other 3 carriers, it doesn't cover more
area, so if you happen to be in an area with bad coverage, your calls are going to be noisy, "go digital" (which, on Verizon or Sprint means they'll sound as if you were under water), etc.
Always choose the carrier first, based on the coverage in the spots you need coverage (and, if you need coverage in an entire city, drive around for a day with your new phone and carrier and check out loads of spots all over the place),
then choose from the phones that will work with that carrier. Choosing the phone first, then choosing a carrier that sells that phone (which is what a lot of people do) leaves a lot of people complaining about the phone when it's really the coverage that's causing the problem.
As for a lot of loud background noise, no phone is designed to handle that. Some hands-free devices (both Bluetooth and wired) handle it better than others, but with devices coming into the market and going off the market so rapidly, you have to find a store that will let you return the device if it doesn't block background noise (Walmart is pretty good with that, and has a pretty good selection, and the same is true of Amazon), so that you can find one that does. My Motorola 850 (Bluetooth) does a pretty decent job, but it hasn't been sold in years. What looks like its replacement (which hasn't been sold in years either), the 720, can pick up a shout from a block away. So you really have to try them. (And don't go by price - I've heard that some of the cheapest Bluetooth headsets, like $12 ones, seem to drop dead on pickup about 2 feet from the headset, but the user's voice sounds, to the other party, as if it was right in his ear.)