Each hemisphere has to know what is happening on the contra lateral side. All the tracts (with a few exceptions) do cross the mid line to reach the opposite side.
For the corticospinal tract in the human that's mostly true (about 85% of fibers decussate); but less so for other descending tracts (rubro/reticulo/vestibulo)...in any case, yes, that's the way it is, but why is it advantageous for each hemisphere to be mostly associated with the contralateral side of the organism? This is one of the great unanswered questions in neuroscience!
could be, but quadrapeds are also crossed...it does seem to track with evolutionary 'newness' though-the more recent tracts are more crossed than the more ancient ones; although that could be a red herring as this also matches medio-lateral control (corticospinal most lateral, vestibulospinal most medial).
To test it you need a mouse that you can specifically block pyramidal decussation; all of the molecular bits were worked out by Tessier-Levigne in the late 90's/early 2000's, so it would be possible to make it!
It could be due to the enormous changes that took place during telencephalization which are directly proportional to the vast amount of new information coming in to the CNS.
Yes, fascinating question. Not only humans have a right/left crossing, but don't all tetrapods & lots of others? First we need to know which animals have this or comparable crossings, e.g. what is the situation in Protostomia? That the right hemisphere controls the left hemi-body & v.v. is very ancient AFAIK (I have never studied this in detail & woud like to know more about it): early Bilateria? One possibility is e.g. that the whole body in a very primitive (bilateral?) worm-like animal was turned left>right (or v.v.) vs the head, e.g. evolving from bottom-feeder to surface-feeder in very shallow waters or so??
In our eyes, the information from the right visual field falls (through the lens) in the left part of each (right & left) retina. That implies an important crossing of information. And of course, left & right parts of the body have to work together, which also requires crossings.
Humans have a lot of direct connections (rel.more than most mammals & even primates?) between our Area 4 (pre-central gyrus: for fine "conscious" control of movements) to the medulla, and about 10 % of these cortico-spinal connections are not crossed IIRC (medical neurological studies 45 years ago...), but remain in their own half. Our very lage brain works of course slower than smaller brains, but the direct connection (without intervening neurons) speeds up the reaction.