A deductive system can come with an answer and a proof of that answer, where proof is whatever counts as proof in that system.
So the notion of “does it really understand it’s answers” gets punted off its Q&A abilities and onto its ability to justify its answers.
What you're describing is what we do at school. We can't assess understanding so we assess justification of answers as well as other things like ability to do X (we don't care if they understood or not, just be capable).
Absolutely. The term "AGI" came about specifically to avoid existing philosophical arguments about "strong AI", "real AI", "synthetic intelligence", etc. Those wanting to discuss "true intelligence", etc. should use those other terms, or define new ones, rather than misuse the term AGI.
AGI requires nothing more (or less!) than a widely-applicable optimisation algorithm. For example, it's easy to argue that a paperclip maximiser isn't "truly intelligent", but that won't stop it smelting your haemoglobin into more paperclips!