oscilloscope performance at slow <1 Hz signals
What's happening in #1 is that the scope is acquiring data on the left hand side of the screen, then switches to real-time (ish) plotting on the right hand side of the screen. 

When you set a trigger point on the oscilloscope, you are basically saying that you want the trigger event to be at the middle of the oscilloscope screen. So, that means the scope has to acquire half of the data before the trigger event occurs. What that means, is that the scope has to have a continuous buffer that has to store info in case a trigger event pops up afterwards. So, it uses a rotational buffer to make sure that it has the data it needs when it needs it.

So, the "dead time" is going to be the duration of the left half of your screen. At 1 second/division on a Keysight DSOX1002A you'll have a delay of 5 seconds while the scope captures buffer data. Then (in auto trigger mode), the scope will automatically trigger and plot out the second half of the data.

2: For the Instek, it sounds like they are simply saying "forget it, we haven't found a valid trigger, grab new data. And, if a valid trigger event occurs we'll maybe catch it next time" The Keysight (and other rotational buffer based oscilloscopes) will give you the trigger where you want it if it happens to pop up in the 1st half of that data collection. Instek won't read it as a trigger (to my knowledge).

3: Tek has an interesting philosophy on this. From a Keysight standpoint, I appreciate that what you see is what actually happens. For example, if I change the V/div setting, I don't want to see old signal on screen that was captured at a different V/div setting. It's a philosophy thing more than anything. In fact, you'll see this for acquisition modes too. For example, if you are in High-res mode and change a V/div or time/div setting, the oscilloscope will actually re-run the high-resolution plotting algorithm on the same data. It's done in hardware, so there's not noticeable delay. Tek, on the other hand, will just keep the old plot because their plotting is a software/processor thing and can take a while under certain circumstances.

I hope that all makes sense! I highly recommend trying roll mode, no delay!