Electronic – Which infrared bulb should I use with a photodiode

There are not good answers to these questions because LEDs are intended for emitting light, and as such the parameters you need to answer your questions are not specified.

A LED reverse biased as a light sensor is a current source proportional to the light level. Being a current source, it have very high impedance (a perfect current source has infinite impedance). The response time is proportional to the resistance of the node times the capacitance. Since the capacitance is parasitic, it is hard to guess and will depend a lot on the particular LED and on layout. The resistance is the deliberate resistance R1 in parallel with any leakage resistance and the resistance of the LED being a imperfect current source. Other then R1, these are again hard to guess. 20 MΩ is so high that leakage can be a important factor. Even dirt on the board and ambient humidity matters at that impedance.

As for how to determine the voltage, that again must be done experimentally. Unless you have a unusual LED that is intended also for reverse operation, you're not going to get a spec. Test a few and leave lots of room for device variation.

I would use a considerably lower resistance with some amplification. The lower resistance will decrease the response time and make things more predictable by making the leakage resistance small enough in comparison to not matter. You are currently getting ouputs from 150 mV to 5 V with 20 MΩ. With 2 MΩ instead, those voltages will be 15 mV to 500 mV, which is still big enough for plenty of opamps to amplify reliably and should be low enough to make leakage ignorable. It may still be too slow, in which case you can use lower resistance still with better amplification.

Another point is that if your supply is large enough to get 5V on R1, then you may be applying too much reverse voltage to the LED in low light conditions. Check the LED datasheet (this usually is specified) and make sure you're not exceeding the reverse voltage limit. A lower resistance will let you use lower reverse bias voltage.

I think I would forget about analog filtering altogether.

Your signal will be sampled digitally, and digitally filtering is a lot easier: sample at a rate high enough not to miss the 100 µs pulse, like at 20 ksps. A full cycle will then have 1 to 3 high level readings from the pulse, and the rest the ambient offset. Average the high level pulses and subtract the average of the low level ones and you're done.

Analog filtering may deteriorate this, since you won't get the nice rectangles with relatively flat level high, and ditto low.

The ideal would be that you can synchronize sampling with the pulses, so that you can do a quick series of samples in the middle of the pulse, which you can then average.