In relation to the domestic violence thing, you have to be CONVICTED for it to affect your firearm rights. As long as you don't have a conviction or currently under indictment then you're good in that department.
To answer your question about what happens after those 3 days, it can vary a little bit. The first thing that happens is that the FBI sends a denial to the FFL. Many FFLs will contact you and let you know. Many will "buy back" the gun from you, but you won't get all of your money back usually since it is a used gun at that point (even if you didn't shoot it, it is still pre-owned and loses value). This is completely 100% up to do the dealer if they will do this. They are under no obligation to do so. They can just throw up their hands and say "Sorry FBI, it isn't my problem. You deal with it.". If things get to the point where the government has to come get the gun from you, you'll be found and will have to answer some fairly serious questions. I can't say exactly how it would go because each case is different. I would venture to guess that someone who was obviously prohibited (like someone with a long violent criminal history for example) would probably be found and arrested immediately. For a person that has a "complicated" history and it might be questionable if they are prohibited or not, my guess is that they would come talk to you, do further research into your history, and then make their decisions on how to proceed from there. If the government has to come collect the gun because you're prohibited, then you're just simply screwed on the money side of things. They will not pay you for the gun, give you a tax credit, or anything like that. You just lose out on the money. If their investigation showed that you were not prohibited, they would return the gun to you.
Typically if you're prohibited then you know it for sure. I'll run over the major points that will prohibit you. I'm not an attorney so don't consider this any type of "official" advice, but it will give you a general idea.
Are you under indictment in any court for any crime that a judge can imprison you for a year or more?This includes any Felony charge or any misdemeanor that has a possible sentence of over a year.
Have you ever been convicted of any crime that a judge could have sentenced you to a year or more in prison?
This includes any felony charge or any misdemeanor that has a possible sentence of over a year.
Are you a fugitive from justice?
This is pretty obvious. Are you wanted? Do you have a warrant for your arrest?
Are you the unlawful user or addicted to any controlled substance?
In other words, do you do drugs?
Have you ever been adjudicated mentally defective?
Have you been involuntarily committed to psychiatric care or claimed "insanity" in any court cases?
Have you been dishonorably discharged by any branch of the Armed Forces?
Marines, Army, Air Force, etc
Are you subject to a current restraining order for harassing, stalking, or threatening your child, intimate partner, or child of such partner?
This does not apply to 3rd parites (such as a friendship gone bad) unless the order specifies no firearms. Intimate partners and children though are an automatic no-go.
Have you ever been convicted of a domestic violence charge?
This applies to any court, in any jurisdiction, for any type of domestic violence charge.
Have you ever renounced your US citizenship?
Are you an illegal alien?