Sounds like either a power issue or a wiring issue that a mechanic should check out
yes
While safely parked
pullover and use the phone only when you are parked in a safe and secure location
yes and yes
My bet it is the spark plug wires or spark plugs or rotor cap if it has one. If the wires are going bad, they are worse "under load" (while driving).
Also, the Toyota Tacoma 4X4 SR5, without Keys, parked, no one in it?
Most comin failure on these cars is the opti-spark distibutor
Depends on if it drained while you were driving or ran down while it was parked overnight. If the battery runs down while you're driving the vehicle then more than likely the alternator is defective. If the battery drains while the vehicle is parked then the problem is either a dead cell in the battery or a light is on somewhere on the vehicle. Could be a under hood light, trunk, glovebox, dome, or even the brake lights. It can also be a relay that is stuck.
Good chance you have a bad head gasket or a cracked head.
While driving - no. While parked - sure, assuming you can find reception though. Look at antennas for RVs and similar to find stuff with the highest chance of working for you.
Blown fuse and/or damaged wiring loom.
What makes you think it only leaks while parked. -