While severe overheating is generally just a cooling system issue...The cooling system on the 2.7 is intertwined with the valve timing system via the internal timing chain driven water pump...
First check for stored codes(looking for cam, crank and misfire codes),
To check codes turn key from off to run 3 times quickly stopping in the run position on the 3rd turn..Odometer display will show stored codes and then "done"..write them down...you should at least have one stored code since you replaced the bat.
Here are the code descriptions:
then check coolant level and oil level...Look carefully for signs of coolant in oil or what looks like overfilled with oil..Don't keep trying to start it and force it to try to run, if it is timing issue you can cause further expensive damage...
Some likely possibilities based on your description of sucking air, loud rubbing noise 5 second shutdown are:
water pump failure cause overheating and timing issues
timing issues cause cam and crank signal not to match = engine shuts down after 5 sec.
Coolant leak, bad radiator fan or stuck thermostat caused overheating and possible warped head
Many other possibilities but start here and post results>>>
Good Luck! and WELCOME to THE site!!