2. Wearing a life jacket is always a good idea, too.
3. Kayaks are not made to carry dogs.
4. If you do take a dog in a kayak, make sure he's tied to you, and it's not a bad idea to put him in a lifejacket as well.
The reason for the skirt is because even the smallest waves splash water into the boat. Once your kayak gets a couple of inches of water in it, it is surprisingly harder to keep steady. If you were to try to get out of the river in a place that is running fast without any still water near the edges, and you had a paddle and a nervous dog in one hand and were grasping on to a slick rock with the other while the current was pushing the back of your kayak perpendicular to the flow of the river, you would very easily get off balance if you had water in your boat, and you just might flip over.
If that were to happen, I would suggest first going for the dog. Toss him gently onto the river bank and then swim like mad for your kayak. If you knew you were not going to make it, jump on to the bank and run like the wind to catch up to your kayak. It doesn't matter if you have to run through thorny weeds. Kayaks are expensive, and if it belongs to a friend, you will feel especially bad if you lose it. If you dive back into the river and still can't catch it, count your blessings if someone else goes after it in their kayak. Don't forget about your dog. Go back and find him, more than likely shivering and hiding in the weeds. Snuggle him, find your friends and pray that you get everything back safely. When you do get everything and everyone back safely, breath a sigh of relief, share your stories, and laugh.
Yes, I think that is what I would do.
This is by far the safest way to kayak...just in case you didn't catch all of those important lessons.