The mesh SSID is broadcast, but normal clients won't be able to see it (FromDS and ToDS are set to '1'). You would need a hacked/modded WLAN driver/card to see it and be able to send packets that way.
Yes, if you have a card/driver to send deauths you could disrupt the link. Capturing the 4 way isn't going to do you much good, esp when using a sufficiently complex key (the mesh is WPA2 AES, so same strenth as WLAN AES WPA2).