First, "not knowing" just means "has more to learn". It doesn't mean "stupid". I was not trying to be rude, but your initial post indicated someone more or less completely lost. It's difficult to teach someone everything through this kind of channel. But now when the fact that you certainly doesn't lack experience is well known, I'm sure we'll sort this out. Sorry if I annoyed you - that wasn't the intention. I was just concerned that you seemed to not understand even rather basic camera handling.
Myself, I've had Canon cameras since 1975 (starting with the Canon EF you can see in my avatar to the left), but I'm still learning things too.
Anyway, it seems that by looking at individual images and conclusions drawn from that, you've misunderstood how this works.
The cameras you refer to, the EOS 1V (and predecessors), EOS 10D and EOS 1DX, all work the same when using flash in P mode. They select a shutter speed in the range 1/60 - sync speed (sync speed varies between models, but is often about 1/250 s). Exactly what you get depends on the ambient light. If dark, longer time and if bright, shorter time.
Using auto ISO when shooting with flash is rather pointless. You don't say you do, but it seems so from reading between the lines. Due to the combination of metering for ambient prior to exposure and metering the flash just before the exposure, the camera has to set a fixed ISO anyway. It's more efficient to select a fixed ISO that works well with the conditions, than relying on having it set to ISO 400 (in P mode it will be lifted to ISO 1600 if you bounce the flash).
The 1DX does indeed remember the chosen aperture from one session in Av until you return to that mode next time. If it was different, then you happened to change it by mistake. Or the camera did, due to being set to safety shift: Av/Tv, but then that's a setting you've done. Since you wrote that you had once a T90, you may remember that back then, it did not remember. Every time you set that camera to Av, you got f/5.6, and each time you selected Tv, you got 1/125 s. Not so any longer.
What's normal for Av mode is to use the full range of exposure times available, which for a 1DX implies 30 s - 1/250 s, when using flash. First the camera meters the ambient light and selects an exposure time that fits the light and your chosen aperture. Then the E-TTL II system meters the flash light and selects a flash power that suits the aperture and the subject. With this combination you get a properly exposed subject, both far away background and nearby parts.
If you set Av mode to use the shutter speed range 1/60 - 1/250 s, you get the same as with P mode, except in Av mode you can still set the aperture. The advantage here is that you never get times too long to handhold (assuming reasonably long lenses), but at the cost of dark backgrounds if it's dark around you.
Locking the exposure time to 1/250 s in Av mode is pretty pointless. It's better to set that in M mode, since that means that you can much easier modify the time, if you feel that you want to, without having to go into the flash setup menu to release the limitation of the shutter speed.
In M mode it's up to you to set how you render the background. The camera's built in metering system will let you see immediately how your setting is, compared to what the camera considers to be a standard exposure. Thus you can easily set how dark or bright you want the background, and still keep an eye on the shutter speed, without taking your eye away from the viewfinder.