Amplifying 40ns pulses that come at unknown times (for single photon detector)

I need an amplifier that can amplify signals at a speed of 1/(5ns) = .2*10^9 Hz = 200 MHz?

Ish, not really. If I squint hard, your pulse looks to have an initial slew rate of 150 V/us, a noise voltage of about 20mV pk, and a time-to-peak of about 8ns. @Kartman's suggestion of a comparator is a reasonable one. High-speed comparators are rated not by maximum frequency, but by propagation delay (with "maximum equivalent input rise time bandwidth" as a secondary parameter). If you can tolerate a delay between pulse start and amplified pulse start of (at a purely wild guess) 10% of 8ns, you're looking at something like the ADCMP55x with 500 ps propagation delay, 510 ps rise time to 90% deflection.

You also ask:

in a perfect world [the channels] will not produce different delays

There is no such thing as exact. You're looking to minimize dispersion. The ADCMP55x claims a max dispersion of 125 ps.

Your noise characteristics are very unclear. If you're lucky, you can simply choose a comparator threshold of something like 80mV and cross your fingers. If you get false positives, then increase this threshold, and/or add hysteresis. Both will increase delay by a configurable amount.

Given that the minimum pulse width for your quTAU is 4ns, assuming that nothing elsewhere in the signal chain has a bug, it is unlikely that a comparator-amplified signal will produce false negatives (your missed trigger shots).

You're using an AC coupled amplifier, which isn't ideal. Buy a 20 dB 500 MHz DC coupled one. Vendors such as minicircuits specialize in parts such as this.

The other option if you cannot find a suitable DC amp would be to buy a higher gain AC coupled amplifier. The high pass filtered pulse should still have the rising edge amplitude that you need provided the gain is high enough.