Seriously? One part of a boxing workout I've never seen questioned before. Pretty obvious what the benefits are. I'm not going to say it can't be lived without but it sure is a major boon to improving one's balance and laying a foundation of good footwork.
Conditions the calves, keeps the feet moving. Aerobic benefits and can be anaerobic depending on intensity.
Nothing much, okay for a warm up and that's it. You can certainly get by without it. I don't buy for a second that it helps with footwork, and if anything it teaches you to lift your legs up instead of pushing into the ground. Because of that I feel like it teaches boxers to avoid sitting down on their punches. You don't box from a skipping stance, and the ground contact times/forces are nothing like throwing a proper punch. Then you add in the repetitive stress on your Achilles' tendon and the tendons in your feet..
Jumping rope isn't meant to simulate punching mechanics, it's meant to simulate movement on your toes in between periods of engagement. You don't spend every second in the ring punching from bell to bell..
That style of boxing isn't really used anymore with the changes in the amateur scoring. Bouncing around on the outside Amir style just makes you look afraid to engage and easy to time. You can move around easily and still be in a position to throw punches. If you're in a skipping stance at any stage you are looking to avoid engaging, and if you do get tagged you have a small base of support so you're going to lose balance easily. To me it's just an old school thing with no purpose other than to show off, like the speed bag.
I thought they also helped strengthen the shoulders / back. Good active rest excercise, I would choose skipping over running.
Gets your hearts and lungs conditioned. Conditions your shoulders and forearms and calves, helps your timing and coordination. IMO one of the most important boxing exercises.