Will the abuser go to jail for having a gun when s/he isn't supposed to?
The abuser can get jail time for having a gun in violation of the law. Under federal law, anyone who owns, has or buys a gun in violation of the federal firearm law can be punished by a fine, jail time for up to 10 years, or both.1 However, as with any criminal case, the actual sentence the abuser gets may depend on a lot of different factors. In addition, realistically, it may be difficult to get federal authorities to prosecute a case. For instance, even though the Bureau of Alcohol, Tobacco, and Firearms (ATF) processed thousands of cases where someone was attempting to illegally buy a firearm in 2015, only 20 of those cases led to prosecution.2
Note: Generally, a person does not have to have knowledge of a specific law in order to be arrested (and convicted) for violating the law. So, if the abuser has a gun or buys a gun in violation of federal law, the abuser can be arrested and convicted, whether or not the abuser knows s/he was in violation of the law. "Ignorance of the law" is no excuse or defense.3
1 18 USC § 924(a)(2)
2 “The Federal Government’s Strange Silence About Gun Crimes,” The New York Times Editorial Board (Dec. 19, 2017)
3 See, for example, United States v. Denis, 297 F.3d. 25 (1st Cir. 2002); United States v. Lippman, 369 F. 3d 1039 (8th Cir. 2004); United States v. Henson, 55 F. Supp. 2d 528 (S.D. W.V. 1999).