Can vCenter 6.5 manage ESXi 6.7?
No, here’s the reference: https://www.vmware.com/resources/compatibility/sim/interop_matrix.php#interop&2=&1=
No, This is not allowed, ESXi can be of same or lower version than vCenter.
Esxi 6.7 can be managed by vc-6.7 and 7.0
Esxi 6.5 can be managed by vc-6.5, vc-6.7 and 7.0
Esxi 6.0 can be managed by vc-6.0 vc-6.5 and vc-6.7