I don't think anyone has answered this question yet. Picard told us in this episode that the tunnels beneath the Chateau were used by the French Resistance in the early years of WWII and then sealed up later in the war, with all kinds of munitions still down there - he cited it as one of the reasons he wasn't supposed to go down there as a child, because it was known to be very unsafe. That's why the gun was there, it was left behind by the resistance during WWII.
As others have said, the older brother was mentioned in an earlier episode. He was away at school on the day of this particular memory - which, although it has been stretched over an entire season and therefore seems like a long period of time, this particular memory is of a single day in Jean Luc's life.
Her nanoprobes were severely depleted after being held captive by the Confederation. She's been working to make more, but they are imperfect and still extremely limited, resulting in imperfect assimilation.
Someone in 1800 might say the same of our society today - they have so many treatments available, why do people still die? I guess the answer is that every single case is a unique individual, people are imperfect and make imperfect choices, even the most wondrous of treatments generally has limits, and in any society people can fall through the cracks. There was an episode of TNG in which Worf was paralysed and wanted to be allowed to commit suicide rather than either live with a disability or consent to a risky treatment, and although he later changed his mind, the episode emphasised his right as an individual to make that choice. By that metric, 24th century society would also allow Yvette the right to choose, so long as she wasn't harming anyone else with her choice.
To be fair, all we know is that the crew believe there is only supposed to be one timeline, and they've been operating off that assumption all season. Just because they believe it, doesn't mean it is true - which is probably the point of Borgati's parting statement.
Regarding the gun in the basement, I must have missed the part where he said there were weapons down there. But it still seems a bit convenient to find one in working order after decades-I'm assuming-of disuse.