If I want to deodorize the carpet with baking soda, should I spray water on the baking soda

The Rug Doctor type of carpet cleaners heat up a relatively large amount of water to a very hot temperature, and spray it fairly evenly into the carpet along with the cleaning solution, then vacuum it back out. Although they are often referred to as "steam cleaners", the rental ones I have used don't actually produce steam, but very hot water (below the boiling point).

A wet vacuum like you indicate will do the last portion of vacuuming up the solution, but the missing piece will be the heating and spraying of the solution. I have definitely had carpet stains that no amount of spray-on cleaner (like "Resolve" spray) would remove, but the rental cleaner would. I believe it is the super hot water that makes the big difference.

Honestly that seems fine to me. For added peace of mind, you can spray it with bleach. If you live in a particularly dry and sunny climate, you could also leave it out in the sun on some concrete. I live in the desert and this works great; the sun dries and disinfects anything in hours.

What you should be more worried about is your drywall. Poke it with something hard. If it's mushy, you'll need to replace the bottom sections. There's also the issue of termites. They're attracted to water; you don't want to find that they're eating your studs. This also depends on where you live, of course.