I had to train my dog to ski politely with me while at the resort, so I can relate. atlanticqueen has great advice. Calm and quiet, anything to downplay the excitement of skiing. It sounds like your dog might have a herding drive, which is a hard instinct to contain.
For me and my high-drive, exciteable dog, it took countless hours and many repetitions. In a word - practice! I took some old track skis with plastic edges, and hiked up hills and skied down with my dog, hundreds of times. I had to train him to trot between my skis, which you probably don't want to do, but if you put a leash and training collar on your dog, and correct him when he gets excited, he will learn. The repetition will also help calm the excitement that he feels when you are skiing. Putting skis on and off a whole bunch of times will help condition the dog to not be so excited, as well (this has not worked for my dog, but we have worked out a deal that he can stay in a down-stay a few feet away and bark all he wants, as long as he doesn't break his stay).
The problem I'm found with the "heel" command while skiing is that dogs tend to trot close to our legs while in heel position. That's how they "keep track" of us and know that they are where they are supposed to be. They can easily get their paws caught under the skis when in heel position, so that's usually not great.
X-C skiing is a little easier, as the dogs tend to learn that the focus is on going somewhere rather than the mode of transport. I am really firmly against alpine/downhill/backcountry skiing with dogs, though. The impact of running down mountains is hard on dogs, floundering through powder is a huge strain on them, and it is almost impossible to keep them from being cut by ski edges. It's kind of unfair to spend a lot of time teaching a dog not to be excited when skiing, and then zoom away from him on alpine skis while he struggles to keep up. I just don't even try, anymore. I have to expect my dog to be calm and obedient at the resort, so it's just counterproductive to get him all jazzed about skiing. I do take him x-c skiing, which is a good compromise for him.
An e-collar can also be a great tool if you spend the time learning to use it correctly. Biting skis and bike tires are terrible and dangerous habits, and the last thing you need is for your dog to bite someone else's tire and cause an accident. Do what you need to do to break those habits - which probably means lots of time and repetition devoted to training.
Good luck! Watch those edges on doggy paws...
. Normally she's a really well-trained, well-behaved dog. Eats when told / sits & stays / doesn't get on the furniture etc etc. Any suggestions ?
