Set them out to dry until complete. Then shelve so they don't get mold.

From experience, yes, if a book is moldy it can spread.

absolutely, yes. here is a quick guide from the nedcc about mold in books and other documents. nedcc is geared towards archival work, but the information is applicable to any document collection.

Wait until they're completely dry to reshelve. Mildew and mold tend to come from persistent dampness (from frequent condensation, drips, high humidity, etc) so if you wait until they're completely dry they won't put the rest of your collection at risk and unless you have other long-term moisture problems in your storage they shouldn't be any more susceptible than any other book.

The spores, man. They spread

Sorry, but your post would be better asked in our Simple Questions thread.