Hi...
DSLR - No. Reason? Kayaking needs the person to be nimble and fast. Lug a weight around not only spoils the fun, but also proves to be rather dangerous, especially when you capsize. It also depends what kind of kayak you are using. If you are using dancers, which are more rounded, you will capsize easily especially when you are new to the sport. But if you are using those fibreglass, rather flat bottom ones, they are rather stable.
Compact, yes but you need to find the right waterproof bag to wrap it. I think a underwatercasing will be bulky. There are good waterproof bags around that can withstand sea-use. Example: I put my mobile using ziplocks (yes...ziplocks and they are good quality ones..)from sheares technology. They are good up to a certain depth..definitely no problem cause I don't see why you will sink to ungodly depths. They are relatively cheap as compared to those bags meant for camera underwater use. Say 1 that can put a compact camera in, probably around 10+ at max. Worried? 2 layers will ensure nothing gets in no matter what.
Where to keep the camera? I used to carry a waterbag to hydrate myself well at sea. Attaching a side pouch to the bag strap (so it ends up somewhere near your chest) will be convenient and makes sure the camera is not in your way at anytime. Or, place the camera carabiner-ed to you between your legs. I presume you will have skirting over the kayak, thus losing it should not be a problem. Carabiner will help you not lose your camera even if you capsize.
lastly, enjoy the sport...since you are new, you may have trouble catching up with others..taking pictures at the wrong time will only give you more problems.
Hope this helps!