They call it a destroyer bit it is actually significantly bigger than a Ticonderoga class cruiser.
Yes.
“Destroyer” is a general utility term these days meaning something like:
“Heavily armed non-aviation surface combatant of variable tonnage.”
The Ticonderoga class cruisers were built on the hull of Adams class destroyers and Ticonderoga was supposed to be the DDG-77 (vice CG-47). I have a booklet from pre-christening that calls it that, in fact. However congress thought it cost to much for a mere destroyer so the Navy said “Destroyer? Who said destroyer? It’s a ... a ... cruiser! yeah, that’s the ticket” and congress paid up.