Allow me to pin bars to other docked bars
This looks like a pretty complicated change, is potentially confusing (in your example, do the tabs conceptually attach to the pane on the left, or the pane on the right? What if you close the associated pane, do the tabs just disappear?), doesn't seem to have that good a use case as it more or less just moves where the tab is, so I don't think it is likely we will make this change.
Allow me to collapse the pinned bar by moving mouse off the bar
This is more or less already how it works. If you click inside the bar to focus it (so the caption lights up), the bar remains visible even if you move the mouse off the bar, as in this case the assumption is you are still working with that bar. However if the bar is not focused, moving the mouse off the bar will collapse it. This is also how it worked in C2.
Allow me to choose the order of the pin tabs
The order is the order in which you pinned the bars. I guess we could add an option to drag the tabs to reorder them though? However we first need a solution for the tab visibility issue (see below).
Allow me to change the text/background colors
I'll look in to doing this, hopefully it will be straightforward.
Limit the size of the tabs
This is an easy change so it's done for the next beta.
Allow me to see all pin tabs on a bar edge
If the pin tabs show on top of the bar that appears, it ends up covering parts of the bar interface in an annoying way - for example it can overlap text making it harder to read, overlap tree expand/collapse buttons so you can't click them, etc. So I don't think the tabs should overlap the bar that appears. In C2 this was solved by reserving space in the user interface for the tabs, effectively indenting the rest of the UI. Perhaps we could do something similar.
Allow me to make the bars expand/collapse twice as fast
The timer is just 0.5 seconds, and feels quick for me. I think you might have been confused by the way focused bars stay visible, as described previously. Typically hover actions like this do not act instantly, or too quickly, as it makes for a fiddly or "slippery" feeling user interface that instantly punishes mistakes like briefly moving the mouse in the wrong position. Construct's context menus use timers when hovering submenus for a similar reason, which are currently set at 0.3 seconds to show and 0.5 seconds to hide, so the pane hide timeout currently matches the submenu hide timeout which seems appropriate. I can't tell exactly but it also feels about the same as the behavior in C2.
Allow me to keep my open bars even after logging out
This is a separate issue really. I'd say the solution is don't log out. If you need to switch accounts frequently for some reason while also keeping your UI intact, which sounds to me like an edge case, you can try using features like browser user profiles to more easily manage that, as you can change account by switching browser profile rather than logging out and back in. If you are being auto-logged-out frequently, it's probably your own browser clearing cookies, so you should probably look at changing your browser settings or making an exception for Construct.