Photon counting circuit
1) You don't need an inverter. AC couple the input to the comparator, and set the detection levels below the new reference point. However, since you do need some gain to get your signal amplitude up, an op amp (quite high speed) is indicated, and you should be aware that you can typically configure an op amp as an inverting or non-inverting amplifier, so you get inverting for free. However, this will depend to some degree on exactly what your PM device output looks like. A voltage regulator is, well, so far off as a consideration that you need to talk to somebody about why you think it MIGHT be appropriate.

2) A 997 is just about right, since it looks like your pulse width is on the order of 50 nsec at half-amplitude. So you need to talk to somebody about why you think a low propagation time will produce too many samples.

Also, you need desperately to talk to somebody about why you are unable to get a sample 997 to work. Until you do, you are dead in the water.

3)The choice between asynchronous and synchronous is pretty much irrelevant, although the problem of ripple-through delay needs to be addressed for an async counter. Since you don't seem to understand what this is, I'd recommend using a synchronous counter. You will need to provide data transfer to a holding register for transmission and a counter reset once per millisecond. Also, you'll need to address metastability issues during your transfer (what happens if you get a pulse while the transfer is occurring?)

4) Speed is not an issue. Assuming you use 2 bytes per packet (16 bits/packet) your data requirements are 16,000 bits/second, which is much lower than the 2 Mbit/sec you can get from Bluetooth. Of course, you're going to have to learn how to handle this stuff.

As you say, you have a lot to learn. I suggest you find somebody who has a clue and start picking his or her brains. That or read a LOT.

Good luck.