Would a transistor be the correct device

About the question 1, it is not necessarily wrong provided that you have a load between its collector and the positive or between its emitter and the negative, and its impedance is high enough to reduce the current through the transistor. That's the main problem here: Q2 is shorting the battery, which will probably result in one of two things: fry the transistor or exhaust the battery (or maybe both).

So my advice: put the LED and the resistor in series with Q2, not in parallel, and use a PNP transistor for Q2 instead of an NPN.

About the question 2, if you want to wire a GPIO to the circuit to replace the switch (i.e. connect it to the 100K resistor), I don't see a problem, the resistor is high enough to limit the current drawn from the board.

It could be a problem if you tried to power an LED directly from the GPIO.

Of course you'll have to remove the switch, because if it is closed when GPIO is high, you'll be shorting the GPIO output to the ground.

I checked the manual of the product you refer to (there IS a manual, however brief). I think I understand what you want and why.

The thingy is a transmitter that is intended to be connected to a normal (mains) switch, and to be paired with a receiver that mimics the switch state at another location.

You want to duplicate the effect of the normal switch with an arduino output, so you can use the arduino to remotely control some mains powered device.

Without knowing what is behind the two pins that must be connected the only circuit that is a 100% replacement for a switch is a relay contact. Assuming the switched current is very low a 4066-style CMOS switch (as Jim suggests) will also work.

Knowing a little about the circuit in the box will likely make the circuit much easier. Very likely either T1 or T2 is connected to either ground or 3V3.

Let's for instance assume that T1 is connected to ground, hence T2 must be pulled low to active the transmitter. In this case a simple diode from T2 to the arduino output will do. If T2 is connected to 3V3 (hence T1 must be pulled high to activate) (assuming you use a 5V arduino) a simple voltage divider between the arduino ouput and T1 will do (1k + 2k2).

The knowledge about T1 and T2's connections can be obtained by using a multimeter, or by inspecting the PCB inside the transmitter.

You circuit might work or not, depending on the polarity of the current that flows from T1 to T2. You circuit assumes that T1 is always more positive that T2, which (without further information) has a probability of 50%.