Carbon Booms:
1. How much does it improve overall speed and performance?
Not at all for shorter booms. Significant for longer booms, say 6.6m and upwards2. Does it make handling any easier, especially through the gybe?
Not because of weight as they are usually almost the same. Yes for larger sail stability by reducing flex and change of boom length. No for small sails, even maxed out 5.0m-5.4m speed sails. 3. Do you notice the slight difference in weight?
No. There is usually NO or insignificant difference in weight in most brands between a good quality alloy and most carbon booms.4. Any rigging improvements?
Yes with bigger sails as they hold the length and shape a little better.5. Does it improve the wind range of each sail?
Yes with bigger sails.6. Will you get greater hang time in the air after a high speed catapult (just joking)
No, you will slam into the water harder as your rig catches the water if you booms don't break and let you fly.
But seriously, I usually use alloy booms on my small speed sails on epic days BECAUSE they WILL break! I have had some serious hurt with a catapult at 40-45 knots when the rig hits the water like a giant sea anchor and stops dead. Something has to give or your body also suffers massive deceleration pain. Since alloy booms are very cheap, I prefer it if they break and not my body or my expensive carbon mast. If the boom does not break you have to hope the harness lines break. If not the harness will be torn apart, and the force required to do that hurts a LOT!
In a really big stack, all the hardware above is totalled! Better than my body!

Carbon Masts
1. How much does it improve overall speed and performance?
Slightly, related to easier handling, higher weight. see 2 below.2. Does it make handling any easier, especially through the gybe?
Yes, because they are lighter.3. Do you notice the slight difference in weight?
Yes, sometimes, quite a bit, especially with larger sails.4. Any rigging improvements?
No, not if the heaver mast has the correct stiffness and bend curve.5. Does it improve the wind range of each sail?
No, not if the heaver mast has the correct stiffness and bend curve, except that with a lighter rig you may have slightly less fatigue.6. Will you get greater hang time in the air after a high speed catapult (just joking)
Extra notes from extensive experience with many different masts:
For SDM masts, the difference between 70-75% Carbon and 100% is not that much in weight in most brands. There is high difference between 30% and 70%!
In SDM masts, I am usually wary of 100%C masts and prefer to use 75-80% of I can. The difference in weight is minimal and the increase in longevity and crash resistance is significant. 100% Carbon mast have very thin walls and can be very brittle. That can get damaged from a seemingly innocent knock and then suddenly snap with no warning later with seemingly little provocation when just sailing along.
That said, I do have and use a couple of longer 100%C SDM masts in my largest sails (460 and 520) and I that them like glass when transporting, handling and rigging.
For RDM masts, 100% is very definitely worth it. They are almost unbreakable, and they are significantly lighter and crisper handling than 60% or 80% RDM's. They work best in sails from 6.6m and down. In larger sails, RDM's can be significantly heavier and have more flex = lost power. Personally, I much prefer SDM's in sails from 7m and up, but some sailors like them up to 7.5m in certain sail models.