Which microcontroller and transistors can I use to control these switches?
It depends a little bit what exactly this switch connects. You for sure can replace it with a transistor but to tell you how we would need to know the schematic of this thing. Otherwise just use an opto-coupler and you're good.

If the switch is used to "connect" a logical signal to an input, figure out how it's connected (could be normally opened or normally closed). Then create a transistor circuit that brings the same signal level to the power input pin when driven.

edit: According to your edit you can simply do something like







simulate this circuit &ndash; Schematic created using CircuitLab

As you mentioned, connect the controller to the same ground potential as the transistor circuit. What controller you use probably depends how you actually wanna control it, generally speaking, an ATiny could do the job. And one transistor per switch is enough.