I have some stp16nf06l and clones of them. They have a gate threshold voltage between 1 and 2.5V, which is fine for 5V Arduinos, but on the limit for 3.3V Arduinos. I have never had a problem with heat, but none of my projects have needed to switch more than about 4 or 5A.
If you want to get a transistor to help your MOSFET switch fully on, any common npn will do, for example bc337.
Another problem that may be contributing to your dim LEDs is using a breadboard. They are not meant to deal with the levels of current you are using. Their spring contacts and internal strips could be adding significant resistance, and also heating up. What I might do is build myself a small module out of stripboard. Solder the MOSFET and screw terminals X 4 (12V in, 12V out, GND, switched-GND) to that and a pair of hook-up wires or PCB pins (GND, MOSFET Gate) back to the breadboard. That way, the high currents don't go through the breadboard. On the parts of the stripboard tracks that carry the high currents, put a good thick layer of solder, enough to cover any holes.
You may also want to make a 12V wire or pin from the mosfet module back to the breadboard, so you can later build a voltage regulator circuit to run the Wemos, and/or for your npn transistor circuit. This will not carry much current, so you can prototype that on breadboard no problem. 12V is ok on breadboard, but currents over 1A are not.