But do they share their mempool, when a node disconnects from the network for some time for example?
When the node goes offline it will not share any information, it will simply be updated as soon as it comes back online.
I think that the mempool size is too large and it would be difficult to share their mempool state, but maybe I'm wrong...
The mempool is quite small as it will only contain unconfirmed transactions (usually a few MB). If a transaction has a too small fee, it will be deleted from the mempool after some days, so the mempool will never be to big.
What you maybe mean is the blockchain, that is a fe hundred GB of data. This will indeed take some days to synchronies if you start your node for the first time.