Not always, not necessarily if the tab was in the background when closed (switch then close), not if they force quit (like the other comment I upvoted says), no

Not always, not necessarily if the tab was in the background when closed (switch then close), not if they force quit (like the other comment I upvoted says), not if chrome updates etc. The other alternatives are opening another type of connection like an RTCPeerConnection to your server and seeing when that closes.