If you see spots or cavities in your baby's teeth, should the baby see a dentist asap?

Yes. . It is a good idea to have a dental exam for an infant when the first teeth arrive in the mouth. Any irregularities noticed (or concerns) after the initial exam should be brought to the dentists attention.
Absolutely. Lots of things can discolor teeth, cavities, stains, too much fluoride, the shape of the tooth enamel itself, etc. 1st dental appointment age 1, or as soon as 1st tooth erupts. Go now, have it checked out and treated if indicated. If it's a problem, it will only get worse. If it's not a problem, you will receive peace of mind.
Fluoride. Depending on where you live it is worth considering the amount of fluorides in the water and what type of vitamins you give him . To much Fluoride can stain teeth. Make sure he has dental visits at least every 6 months so they can evaluate for other things.
Yes. For cavities yes. Spots might just be a stain from food. For cavities i would see the dentist as they might be able to help correct whatever problem is causing the cavity in the first place (and prevent more from forming).