- Should I wire the power supply to the LED strip directly or should I use the power jack on the Arduino UNO directly?
You should wire it directly to the LED strip. The LED strip and the power supply form the "real" circuit. Whereas the arduino will be supplying only a low power digital control signal.
- If I use power jack on the Arduino UNO what will happen if the LEDs are powered on and the external power supply is disconnected before
The LEDs will draw excessive power from the USB port. The port would probably shut off due to internal protection. I don't think anything would be permanently damaged, but it's possible. Regardless, you are right to want to avoid this situation.
- What do I need to do to protect the signal pins on the Arduino UNO?
It's actually pretty simple. The arduino and the LED strip need to have their grounds connected, but they don't need to share a power supply.
First connect the LED strip directly to the 10A power supply with decent
gauge wires, like 18-22AWG. This is the "power" part of the circuit.
Connect your arduino to the LED strip using 2 wires. GND, and SIGNAL. You can add a series resistor to the signal line for additional protection. A 1k will probably work.
Now one additional thing you need to look out for is ground loops. The ardino and led strip are connected by a tiny ground wire, and if the ground potentials of those 2 systems are slightly different, it will cause problems (probably unreliable communication). The simplest way to prevent this is to plug the power supply and the PC into the same outlet or power strip.
If you can't do that, or it's not sufficient, you can add an opto-isolator between the two signals, but don't worry about that until you've tried everythign else.