You can check your heater core by splicing the 2 heater hose, that willl bypass the heater, if it still acts bad then not the problem.

The weep hole on the pump is on the top in a well. The usual failure is the volute come loose from the shaft (no noise). This is not an old fashioned water pump.

This is one of many on ebay $59.68 The part designation is 2850, that is a fairly standard number.
https://www.ebay.com/itm/2850-New-Ra...rd!22407!US!-1

 

Not trying to sound like an idiot , but would the heater core make it run hot ?

 

Could cause a sort of air pocket. There is supposed to be a constant flow thru it on the hot side of the t-stat. It flows from the head to the t-stat housing, so cavitation could occur.