First of all, why does Bisq care about Bitcoin Core being installed or not?
Then why does it require it to be fully synced and not pruned?
Finally, the error message talking about "Bloom filters" is confusing me, what are they and why does Bisq need them ?
Bloom filters are a legacy way for a light client to query transaction data from a P2P node. They are defined by BIP37 and we usually refer to these clients as "SPV wallets".
Why does Bisq care about Bitcoin Core being installed or not?
So that it can query transactions from localhost instead of scanning from some external source (with the well-known privacy leaks of bip-0037).
Why does it matter if it's "fully synced" or not before Bisq is started?
Considering the application, they may want to be sure to be to the latest Bitcoin state before taking any action on their own P2P network.
Why does it matter if it's "running in pruned mode"?
You'd be missing blocks and potentially some historical transaction data that Bisq might want to query for.
