"Does it belong to the Queen as part of her personal estate, or does it belong to the nation and is used by the monarch as an official residence?"
Belongs to the nation, and much if it is offices.
'Don't they have a large art collection, which could be put on display, and could raise money?'
Yes, they're out out on loan for display regularly. But not at profit because they're already owned by the nation.
"It doesn't need to be a workplace for the monarch nor an official building for their events because there's no need for a monarchy."
Yes, there are other models for arranging a society, but this is the one we have. And for as long as there is a monarch, there is need for a workplace just like any head of state.
"Why not sell Kensington Palace and turn it into luxury apartments (they would go for serious money!)"
It already is luxury apartments! with pretty normal market rents. The Monarch paying from own pocket, not official funds any top ups required for the tenants to afford it. Which seems a bit left hand - right hand, but transfers a fair amount from personal to official funds.
"If it's in such a state, knock it down."
Splendid idea, I think it's ugly. How much would a rebuild of something more modern cost though? (more, probably)
And I think the lead needs to be taken by Parliament. Because the estimated cost of fixing that isn't like this - a rounding error on the NHS budget, it's about 3 years worth (if higher estimates are nor scaremongering)