It's $125 and you have to call Delta.

If it's flat bed seating you won't be allow to have the pet and it can't be a bulkhead since the carrier has to go under your seat in front of you.

Even though the animal is suppose to count as your carry on, I've never had any issues bringing a carry on roller bag and my backpack on with me when travelling with cats but I'm sure others haven't had as good of luck. I think the policy is stupid seeing you are paying a high fee to travel with the pet anyway and if someone is with their pet the last thing they want to do is spend more time checking a bag and retrieving it after.

I'd definitely use a soft side carrier instead of a hard plastic one.

You are eligible for upgrades with a pet as well if you don't want to buy a first class ticket, only thing is you can't sit in an exit row.

I've had a couple people complain when flying with cats in First Class claiming cat allergies so you may have that with a dog. Both cases the other passenger had a complimentary upgrade. One time I was on a paid first class ticket and the FA or GA said since they were on a complimentary upgrade and I was on a paid first they would be happy to move the next person on the upgrade list forward and give them their seat and the other time I was on a complimentary upgrade and I was a diamond and the other passenger was a lower tier and they offered to move the passenger to C+ or Y. Amazing in both cases they were able to cope with their allergies in first instead of moving back to coach