I think you're still missing a couple of points. First, "Light-Sport Aircraft": the term means two different things, which causes many people confusion. Some aircraft may be registered as a LSA, "special LSA" or "experimental LSA." S-LSA is a factory built aircraft and certificated as such. E-LSA can be one of three things: either an approved E-LSA kit, a former ultralight that was registered during the transition period (no longer possible), or a certificated S-LSA that had the registration converted to E-LSA by the owner. None of these apply to you Today the only path to LSA certification is factory built (SLSA) or approved kit (ELSA). There are no exceptions.
However, an aircraft registered in another category (standard, experimental, primary, etc.) that meets the LSA requirements (2 seat, 1320# gross weight, etc.) may be flown by a Sport Pilot as if it was an LSA, but it's still not registered as LSA; it's a standard or experimental or whatever. From an operational standpoint it's the same, but the paperwork and inspection requirements are different.
A homebuilt aircraft not built from an approved ELSA kit gets an Experimental-Amateur Built airworthiness certificate. There are certain requirements for this, the most important one is that the aircraft must have been built for "education or recreation", that the "major portion" of the aircraft is indeed amateur built (the so-called "51% rule"). At the time of initial inspection, the applicant is required to show documentation (photos and build log) to prove that the major portion was indeed amateur built. If you have this documentation, no problem, even if the plane was previously flying as an ultralight. If not, then you can't do it, though some DARs may be more picky than others about the completeness of the documentation. I do know of already flying Quicksilvers that have successfully been registered as E-AB, but I don't know what level of documentation was provided.
If the aircraft cannot be registered E-AB (insufficient build documentation) it may be possible to register it as Experimental-Exhibition. This is more restrictive, requiring yearly notification to the FAA of where and when you expect to fly it (airshows and such), though local "practice flying" is permitted. Some people take this route and some abuse it, though I don't know to what level the restrictions are enforced.
In the case of a damaged aircraft, that's an entirely different thing as it already has an airworthiness certificate so it's just a matter of making repairs.