Never run WITHOUT the thermostat >.<

You are having issues with a new thermostat because you aren't properly bleeding air from the system, google up how to "burp" a G35, it's a fairly complex process, if you break the plastic purge valve on the heater core line you can order up a metal replacement one (or just plan on replacing it because it's a MUCH better part anyways and only $25) from Z1 Motorsports.

Air gets trapped behind the thermostat and needs to be purged or the thermostat doesn't open and it starts to overheat, not having the thermostat means no air gets trapped behind however you have MUCH longer initial heat times (excess bearing wear due to slop expansion tolerances for a longer duration on every cold start)...

The whole point of the thermostat is to trap heat INTO the engine so it gets up to temp quicker, not to cool the engine when it's hot.