Is there any harm using a solid fitting when connecting heater hoses together to bypass heater core?
No, it is commonly done when heater core goes south.
Which hose gets cut and spliced back together on a 1998 Lincoln Navigator to bypass the front heater coil?
yes. You would need to bypass the heater core in order for your coolant to still circulate properly. Check with your locat auto parts store. Some stores actually sell a bypass hose designed for this application. If not, you just need to connect your inlet and outlet hoses at the heater core together so it will circulate. Most auto parts stores will have a barbed fitting to connect them.