The last episode aired on BBC Two was, fittingly enough, the Season 11 finale, Behind the Laughter, on Friday May 7th, 2004, I know there was a mini hiatus before that: all in all, it was almost six months until Channel 4 started airing Seasons 2 and 12.
I remember an article that said the BBC held the rights to the series until 2006, even though they stopped airing it two years after 4 got the rights in 2002. In 2005, Channel 4 gave them the rights to Family Guy for BBC Two, and later BBC Three, which is how Seth McFarlane's other shows made it to terrestrial in the UK.
Not sure about that, I'm sure C4 hadn't shown Family Guy for many years when it started on BBC2. As far as I know, BBC2 bought the rights to American Dad, which they showed for the first time on British TV, and Family Guy got bundled in within it (much like how ITV2 have them all now). At the time American Dad got the lion's share of the promotion and they were shown back to back on Saturday nights, for a few weeks at least - https://genome.ch.bbc.co.uk/schedules/bbctwo/england/2005-10-22#at-22.05
As you say, the last episode of The Simpsons on BBC2 was in May 2004, that was a repeat, and the last new episode was in January of that year. From around early 2003, BBC2 would often take The Simpsons off for a week or two to try and launch something else to potentially replace it, not that they ever really found anything, so for the last few months they were really rationing it.
BBC 2 for a while also gave The Simpsons a primetime slot in 1998.
Yes, although that was pretty much only there as a spoiler for the appalling ITV sitcom Babes In The Wood. They showed about half a dozen episodes in the 9pm slot but seemingly it didn't rate any better there than it did at 6pm, so they just kept it there. It was presumably also easier to show 22-minute shows at 6pm where there were loads more imports and less requirement to shove in filler shows to fill up a half hour slot. As mentioned, C4 also seemed to find 6pm did just as well as 9pm.