FWIW, as an example, here's what I am able to capture with a Rigol DHO804 tricked into thinking that it's a DHO924. Cost $342.16 at the time of purchase, but I did use some Aliexpress sale coupons active at that time.
1. A ~10 nanosecond pulse:
2. The shortest pulse that it can capture
and detect its true amplitude (of course including the overshoot), meaning there's at least one sample point present at the peak:
3. About the shortest pulse that it is able to trigger on; here the displayed amplitude is below the true value (vertical scale is the same, but I raised voltage of the signal source to output ~5 V):
(I am not sure whether I hit the limitations of the scope or of the pulse generator here, but naturally I would expect that it's the scope, because it's clearly about the spot where 1.25 Gsa/s becomes the limiting factor)
As you can see, it has no issue triggering on a 1 kHz repetition rate. As a matter of fact, it happily triggers on a 0.1 Hz signal:
All of the above was captured using a 100:1 probe, so ~25 mV on the scope input. It can trigger on lower values just fine, but I haven't tested how low it can go when the pulses are very short.
I also haven't tested it on capturing two consecutive short pulses, but really there's no reason for it to be able to capture one, but not two.
A caveat: this was captured with a single channel being active. When you switch on another one, sampling rate is halved, and the results become accordingly worse.
Now, I'm not saying that it's necessarily a good idea to get, for scientific work, a scope like this, for which the observations you want to make are not very much above the limits of its capabilities, but I do want to point out that a multi-thousand dollar scope is not necessarily a strict requirement. I think it's the probes that you'll need to spend most of the money on, at least as soon as it comes to high voltage and/or differential probing.