Many beautiful boxers should not be bred.
A friend has a stunningly gorgeous boxer from impeccable, health tested bloodlines that is neutered. She is not a professional breeder so he is a neutered pet for his health benefits.
Breeding is definitely more than picking two "great" looking boxers and producing a litter. It's also more than counting championships. This is often what backyard breeders do, breed their "good" looking dog with one with a few champions and they have a litter. Sometimes they get lucky, other times they begin producing a majority of boxers who don't match the standard or have health problems.
I have learned a lot on this site and links from the breeders on this site. I have picked up a lot of knowledge from working rescue and learning about the byb's - some we learn way too much about, and I learn at the dog shows. Sounds like you have many questions and are serious about the boxer breed. I am sure you will enjoy many of these links also.
I hope the above came across in a "friendly" way, so many times when you type info without a discussion it comes across very "telling" instead of conversational. The only way to learn is to come on here and discuss it.

But, I must add, vets are not judges and versed in the standards of all breeds, I would not take a vet's judging of a dog. I would definitely want to find out from a judge's standpoint after going to a few shows for a trustworthy opinion. I know two people locally who were told by their vet to breed their boxers! I am sure your's is very nice looking, but let me tell you these two people have the most awfully bred boxers! I was horrified that they were told to breed! Especially when the world we live in is overrun with pets. The one boxer could very, very well be a mix even!
Sounds like you have a excellent family member in this wonderful boxer. But I agree with all the earlier postings, it sounds like there are too many reasons not to breed this particular boxer. Maybe after you show a bit you will have interest in breeding in the future.
Edit: I just read the links from the above boxerworld post link. They say it much better! I enjoyed the article on linebreeding.
Good luck!
Ashlie