Is it safe to directly use the output of the battery cell pack inside of a portable charger?
I say "probably" 6 times below. This is because one can never be certain what an unknown manufacturer/designer has done BUT I'd expect my answers would be correct in most cases. YMMV. 


  Is this safe to use the battery cell output directly 


"Safe" is in the mind, practice and ability of the beholder.
Yes, it's potentially safe, if you connect competently, do not short or discharge with too high a current or to too low a voltage.


  or is there any safety circuit I missed that would have to be added? 


You need to tell us the cell brand and model.
As the pack is rated at 156 Watts max then Ibattery max is in the order of I = W/V 156 W/(7 x 3.6) = 6.2 A (probably).  If you limit current draw to under say 7A you are probably within battery specifications. 

The cells hopefully have internal cell protection circuitry -  but maybe not. Adding a gross overcurrent protector to the output "would do no harm".
If direct discharging you should never discharge a 7SxP pack to under 3V x 7 = 21 V and ideally a little higher. 


  ... is this going to severely reduce the life of the battery/how could I avoid that) 


As above. Treat the battery like an LiIon battery likes to be treated.


  Could I use the battery from my new port while it's being charged? 


Yes, but. It MAY have a "gas gauge" circuit to help judge capacity (but, probably not) and direct access would scramble that. But odds are it would be OK.
The charging termination may be somewhat confused by direct access discharge, but, probably not badly so.


  Could I use both my new port as well as an output on the existing PCB (for example, plugging a phone in to the original USB output and using the new raw output simultaneously)? Would the charge indicator still be accurate?


Probably. Again, IF gas gauging is used it could get confused, but it most likely has not got that degree of sophistication. Remember that total watts is direct but usual loads. 

The charge indicator is probably based on cell voltage and a little magic, so direct access not a problem.