Yes. Direct contact of baby's skin with the wart will make transmission more likely.
Contagious. Warts are virus cells caught in the dermis layer of the body and contagious. Your immune system was not able to fight it off. Your baby is highly sensitive and the immune system is not ready for these encounters. Please get it treated and once it has healed you can resume bathing with your baby.
Depends. Do you also have plantar's wart? If you don't have a plantar's wart, it's likely impossible.
Yes. As far as is known, no you will not. Now as for kissing a frog.......? ( just kidding).
Yes... It may show up in other places than just the bottom of his feet too.
No. Yes, it is "possible" to give a 7 month baby a plantar wart, by transmission via the bath water from the parent's wart . . . But the chances of that seem extremely rare. Plantar warts form on the bottom of the feet, perhaps when there is roughed or cracked skin, and a person steps on a live virus; but little babies don't walk. It is not known if the virus can stick to a baby in warm soapy water.