As to HST, I believe it would take an up/down situation to cause it to suck air.
As to engine, depends on where the sump and pickup is. Many, many times I've had my offroad buggy at such an angle that it loses oil pressure. First indication is valve train noise, hydraulic lifters losing prime, etc. In all engines the crankshaft is fed first. So chances of hurting bottom end is extremely low.
I'm in steep timber a lot with my tractors. I'm not man enough to ride my Kubota on any slope that would cause it to lose hydraulic pressure or engine oil pressure.
As mentioned before, it's easy to overrate a slope. In the electronic world we live in you can put an app on your phone that can be used to measure slope. I use AngleMeterPro. I won't ride a side slope more than 18 degrees with my Ford which is very stable.
I also downloaded a conversion chart that can be used to determine slope by using a straight edge such as a long 2x4 and a tape measure. If mowing a side slope, 5 to 1 is comfortable. 4 to 1 is doable. 3 to 1 is dangerous.
I've saw several pictures here on TBN of slopes claimed to be some high degree number. If the pic is shot along the face of the slope and there are trees in the pic they can also be used to figure the slope. Rarely is it as steep as claimed. Rockcrawler's pic is a doable slope for mowing. At the far left you can see a couple trees that indicate the degree of slope, as well as the roofline in the background.
