1. You get breeding quality hedgehogs from a reputable breeder. They'll come with pedigrees (4 or 5 generations is standard). I suggest having a pedigree software - personally I use PexEd by Breedmate, but there are one or two others out there, though from my understanding they have fewer capabilities. My method is to enter any pedigrees into the database that I can get hold of, so I essentially have a huge web of hedgehog relations that I can use for reference. Most of the time when I get a new pedigree, after I input the 4-5 generations into the system, I actually know their background going back 10-15 generations or more. Pedigree on its own isn't enough, you need to know what to look for. There's no website to look names up on, but you can contact the IHR and send the pedigrees to be inspected. They'll let you know if there's anything questionable or any names they recognize that you should avoid. You should also have them looked at by a breeder who knows what to look for, because the IHR only has information on registered hedgehogs, so it's best to use two sources to get the most complete opinion. I have, multiple times, seen WHS names in pedigrees. I was lucky with my mentor and the information she was able to give me, but most new breeders are not going to be able to recognize WHS names at a glance, so you need to have the peds looked at by someone who has experience with that.
2. The best method to tell how closely two hedgehogs are related is the inbreeding coefficient. Any pedigree software worth its price tag is going to be able to calculate that for you. This is part of why it's good to know more than 5 generations. The inbreeding coefficient is going to be a percentage. Anything over 6% is considered inbred, but most breeders aim for much lower than that. The lower the better, really. I keep a spreadsheet with males along one axis, females on the other, and all of their COI% kept track of there. Any pairings that will produce offspring over 3% I won't do, but the majority of pairings I choose fall under 1%. I would be willing, for instance, to do a pairing in the 2-3% range if I have a specific reason, such as being interested in a color outcome, and under 3% is distant enough that there won't be any problems from it. But, it's good to be in the habit of going as low as possible. Hedgehogs are no longer able to be imported from the wild, so what we have in North America is what we have - it's very difficult to find a pairing with absolutely no relation (although if you only know 5 gens going back, it might appear to be the case, when in fact they're really somewhere under 1%, but not a perfect 0%). For comparison, most purebred show-line dogs are around 10-12%. A pairing of father x daughter or full siblings is going to be about 25%. Daughter x grandfather is going to be around 12-13%. And so on.
3. When you decide to apply for a USDA license, you can send in a request for the application material. This will include a booklet that outlines all the information and requirements. For hedgehogs, it's fairly straight-forward. Things like proper ventilation, upkeep. Cages have to be clearly marked with the individual's information. Food containers have to be sealed and clearly marked. You have to keep track of paperwork, that sort of thing.
4. There aren't breeding bins the way that there are breeding container for snakes. Large sterilite bins are a cage type a lot of breeders use because they're the cheapest option for that amount of space, and they're very easy to transport and clean, and you can see into them easily, etc. Some other breeders use other cage types, it depends on personal preference. When the breeding actually takes place, sometimes the male or female is moved into the cage with the other, but it's better to put them both into a separate cage because that way neither of them feels like their space is being invaded. Most breeders then use the females cage as the one where the babies are birthed and raised. If they're not already using substrate bedding for all of their hedgehogs, the mother is put on substrate about a week before she's due, because it's better for nesting purposes and for reducing odor (since you can't clean the cage until the babies are weaned). I usually use liners and bags, so when the females are put on substrate, I switch those out for a nest box. Some people use igloos for the nesting area. The details are personal preference.
5. 1. Request application materials. 2. Make sure everything is within the guidelines (touched on in #3). 3. Make an appointment for a vet to come do a home inspection. They have to go over some things with you and mark them off on a list, then fill out some paperwork. 4. You keep the vet paperwork with you and send in the rest of the application. There's a small application fee. 5. A USDA inspection agent contacts you to make an appointment for the first inspection. 6. Inspection happens. You get approved. 7. They email you a piece of paperwork saying you've been approved. You confirm that it's been received. 8. You get a confirmation in the mail, including instructions for paying the appropriate license fee. You send the license fee. 9. When the license fee is received and processed, they send you the approval materials, including a certificate stating your license number. 10. You then have (roughly) annual, surprise inspections.