Input 5V 100mA, output 200mA

Very difficult to say why this is happening without more information (scope picture, PCB layout, schematic etc)
What frequency is the signal? Where exactly is it coming from? (the FPGA? or discrete flip flop - not entirely clear from your question) How long is the trace from the pin? What is connected to it?

It could be that it's the way you are testing things is the problem. Have you removed the probe ground clip, wrapped a short wire round the barrel and attached this to the nearest ground point to the signal pin?
If it is really there, then it could be caused by not so great layout of the PCB. Do you have solid ground/power planes on the board? Is every power pin properly decoupled with the values recommended by the manufacturer?
If you can provide some more info it will be easier to suggest where the problem lies.

Microphones don't put out much signal. You basically need a amp that has a voltage gain of a few 1000 and a low output impedance, like 8 ohms or less. You probably design the amp with 5000 voltage gain, then the volume control reduces that as necessary. A volume control is basically a variable attenuator.

To make is switchable, just put a SPST switch in series with the power. No power, no output.

Multiple signals can be summed in various ways. The cleanest is probably a "summing amplifier", which is just a inverting amp with one input resistor for each input source. This has the advantage that each source sees a resistive load to ground. One source can'b back drive another, although that may not be a big deal if the sources are buffered for that purpose.

The design of a good audio amp could fill a book. I have to run now and am not going to write a book anyway. Look around for battery operated audio amp, microphone amp, headphone amp, etc. I'm sure there are lots and lots of circuits out there and maybe even whole devices. Mic amps and headphone amps aren't all that unusual. There are also small power amp modules that will be easy to use for the final stage if you want to roll your own.

How about something like a MP3 player to drive the headphones. I'm guessing that those or at least some of those come with a line in or mic input. Look around. Good chunks of what you want should be available off the shelf, especially if you're willing to do a little repurposing.