I'll attempt some answer, but must admit it's a bit speculative. I think, like with many unusual/rare phenomena, there's a fair bit of observational bias and of psychology involved as well.
I remember reading about this in at least two popular meteorological publications, one book about extreme and strange weather and an article in (I think) in "Weather", the journal of the UK Meteorological Society. Unfortunately I don't have the precise source and have to work from memory.
My impression was that it's generally accepted that tornadoes and similar phenomena, which are known to lift and carry water and debris, can in principle sometimes pick up small animals as well, which then might fall at a certain distance when the tornado dissolves. One study shows that small items can be carried up to 355km far in tornadoes.
The tornado doesn't just stop suddenly, but gets weaker, so debris and possible animals wouldn't all fall at the same time like a huge swarm, but a few at a time in different locations, and it would only happen in rare circumstances.
There are also - as you also found - many historical reports. Many are collected by Forteans (going back to Charles Fort, who dedicated his life to collecting unusual observations), but are of varying quality, usually from casual observers, not scientists.
With such casual reports of unusual observations, often from newspaper and second/third hand, you have to consider that the story changes as it's told and may not be reported exactly as it happened. So, 2 or 3 fishes may become a whole swarm, and it may be reported that people saw them falling when in fact they just found some dead fish on dry ground and couldn't immediately work out where they came from.
That there are various effects that change peoples' perception of unsual phenomena, and how they are subsequently reported, is better established for other, more frequent sightings like ghosts or UFOs, and of course eyewitness reports of crimes, where there is huge body of psychological research. Basically, what people report (and sincerely believe to be a true observation) is often not exactly what happened, and media reports are even less reliable.
So, in some instances the "falling from the sky" may not actually be observed but an an explanation that people came up with after finding the animals. There are other reasons why aquatic animals can occur on land sometimes in large numbers. Frogs and toads migrate at spawning time in huge numbers, and also toads come out of the ground after a rain. Large number of fish can end up on dry ground when a river floods and recedes. I've seen it myself as a child when visiting relatives in the countryside that a field was full of dead fish and wondered where they had come from, but my relatives told me the area had been flooded at spring melt. It was not at all obvious to me that a rather small stream could flood so much, so I can well imagine that somebody might conclude that they must have fallen from the sky.
So it's not so easy to establish how frequently it really happens that animals fall from the sky. While (unlike ghosts) it's not scientifically impossible and a credible mechanism (tornadoes) exists, it's rare enough that we don't have really good, comprehensive observations.
Now for your questions:
Why one type of animal?
It seems generally fish and frogs. I think there are two reasons, one meteorological and one psychological.
A tornado has to pick up the animals in the first place. These would be have to be small animals. Now, the density of fish and frogs in a pond is often very high, so if a tornado moves over a pond or lake sucks up water, there is a chance that it would lift hundreds or thousands of fish or frogs, so it is much more likely, when thy fall, that they are seen or found.
Small land animals, like mice, tend not to live in such dense populations, and also run away or bury themselves in the ground. So while some will certainly be sucked up in a tornadoe, it's much less likely and probably not many.
The second factor is psychological. Many observations are not actually of the animal falling down, but of animals lying on the ground that shouldn't be there. So, if there's a fish or frog lying in dry ground after a weather event, people would notice and report it. If they find a dead mouse, they wouldn't find it unusual and forget about it.
Also, mice are good at falling from height, so they may survive and just run away and never be seen, whereas a fish on land will just die and start to stink after a while (and therefore be noticed...).
Why no videos?
There's an observational problem here. Ideally a scientist would like to observe first the animals being sucked up, and then falling. Now, tornadoes are not that easy to observe at close range, so observing animals being sucked in is technically very difficult.
Observing when the animals fall is in principle possible, but probably extremely unlikely. First of all, the event itself is unlikely (see above). Then, when it happens, more frequently it's probably just one fish suddenly coming down unexpectedly, not a longer "rain" of fish. So unless you were already filming for some other reason and catch it by chance, you wouldn't have time to get out the camera and actually film it falling.