LED not blinking when using a transistor
Without a common reference (ground) between the Arduino power supply and the external power-supply, again, you have a floating base, as you can't know what the Arduino's 5v output (or local ground) looks like to the external circuit.

Connect the two grounds. That should correct the LED brightness with respect to the Arduino output pin level. Then, if you need the external circuit to behave in a specific way when the base is disconnected from the Arduino, add a 10K resistor from the transistor base to the external +5 if you want the LED to be on, or to ground, if you want it to be off. Again, I'm guessing at both the pullup/pulldown value (but 10K should be in the ballpark) and the switching sense ('On' or 'Off' for base at +5), but from the expectation state in your point (2.), I think I have it right.

Update: But see @EdgarBonet's comment to me about BJT's. I'd still avoid leaving an input floating as a matter of practice, but it seems not relevant in your circuit.