1. My biggest complaint is when panning an aircraft across the sky, the sky goes from grey to blue and sometimes back again. I know some of this has to do with the angle of the sun, refraction, etc, but will shooting in M help any?
Yes. It also helps when shooting a navy blue AC vs. a silver one.
3. Do my settings need to change when changing lenses?
Not as long as you're not shooting at f/2.8 & you put a f/4+ lens on.
I know there is a huge exposure difference between a kit lens and any decent lens,
No. Not as long as the aperture difference isn't what I said above.
4. How often do I need to recheck/change my settings throughout the afternoon?
How often is the light changing? (Sun vs clouds.)
5. Is it okay to shoot a grey card, hand or whatever inside the FL of the lens? My arm isn't long enough to hold anything in front of my 300.
Try this: Need an exposure crutch?
And, if you must use a gray card, just drop it on the ground. It doesn't need to be in focus.
6. Does it matter what metering mode you use when shooting a grey card?
No.
7. Does cloudy vs sunny make any difference in how the process works? What do you do if it's partly cloudy and the sun comes and goes? That could be a nightmare in M.
I address that in the link I gave you. Change the ISO, or anticipate & choose your shots.
Good luck, & don't forget:
Airshow guys - prop freeze?
P-51's, it's all about the prop blur!!
Helicopter Prop Blur
Dramatic light when you can get it, & thinking ahead & planning your foreground & background made for some excellent images here:
Pensacola NAS 2009 Twilight Airshow
Smaller air shows might allow you to get closer so you don't need as long a lens, plus sometimes they also have evening shows: Night Air Show - Alma, MI
Night Air Show, part #2. Aircraft & Fireworks... lotsa' fun?
EDIT-1: While I usually only shoot on manual, I take precautions at events like air shows when I might be shooting a jet on M & a prop job comes by unexpectedly. So I've also set a slow shutter speed on Tv & a wide open aperture on Av, along with the Exposure Compensation (EC) that I think will work at the time I set it up. Then it's just a quick change of the dial to get something usable.
This also covers me on those few times that I accidentally bump the mode dial & don't notice it.
Someone will ask, "Why do you usually shoot on manual?"
Look at Post #47 in the link I posted above: Need an exposure crutch?
EDIT-2: Someone else will come back with, "I've seen good shots that were taken on "Auto". Why can't I?"
Well, you can. Post #47 in the link above will show you that you're going to have to PP the shots where Auto exposures were off & you're probably going to have to delete a bunch that didn't make it, though. I guarantee that some of those will have been shots that might otherwise qualify as wallhangers.
tgara had pretty good luck his first time out here: USAF Thunderbirds
The venue was very bright, but with moving clouds that blocked the sun from time to time. Keeping that in mind, I used auto white balance, ISO 100, and Program Mode. To take advantage of the 6+ fps of the 40D, I shot JPGs, and set the drive to high speed with servo AF. I was also using a Picture Style setting with punched up color saturation, sharpening, and contrast. With those settings, it was fairly easy.... just focus on the planes, pan and fire away.
And maybe someone could tell me how they'd shoot some of these on Av or Tv? Night Air Show - Alma, MI