When saying pinned tabs, you mean shortcuts, right?
I have added these suggestions to the to-do list and should come out in the next update!
Yes, I did not know what they were called.
Currently, you can't. Chrome doesn't have a built-in API for getting these, but you can have a bookmarks widget.
What about Firefox? Which is what I am using.
After some research, it doesn't seem like Firefox supports it.