It looks like both of these issues are caused by the same general behavior. The tool is designed such that when you click a new verse, it refreshes, putting that verse at the top of the page and putting the cursor at the beginning of that verse. This causes the sudden change of positions. And if you use the right and left arrow keys and cross into a new verse, it waits until you pause briefly before performing this same sort of refresh. This same behavior happens whether or not you’re in 2-pane view.
I’m guessing that one reason this was seen as desirable is that usually this tool is used with 2-pane view, and you generally want both panes to line up, in order to make comparisons.
But it seems reasonable to me to request an additional toggle in the menu which would let you turn off the behavior which causes the selected verse to jump to the top of the page, which differs from normal PT windows.
As you implied, it would be nice if the cursor position was not forgotten when this refresh happens. A related quality of life feature would be to delay the jumping behavior (refreshing newly selected verses to the top of the page) while highlighting text. Currently, if you try to highlight text in a verse which is not the current verse, the refreshing behavior causes the window to shift, and your highlighting is ignored and the cursor moves to the beginning of the selected verse. If the refreshing behavior was delayed until after the selection was made, and if the cursor retained your selection, you could use that to reduce the time it takes you to figure out where you were looking before the jump happened.