A solid color stain isn't like paint; it is thin in its consistency and doesn?t form a thick film like paint. Because of this, the stain will penetrate were there is raw wood plus it won't cover the existing paint color very well, this depends on the colors.
Applying a solid color stain over paint isn't a good idea and will probably make you work even harder in the long run. Plus a stain, solid color or not, will not cover caulking in one pass. You will end up applying 2 or more coats of stain to achieve a good look. You might as well prime and paint, it is easier.
All loose paint has to be removed prior to painting and a stain cannot change this. Since the proper prep must be done anyway, why not prime the wood and paint it with a good paint.
As for the deck, careful sanding is the best way to prepare a deck for staining but this is expensive. Another way is to pressure wash the deck using an injectable wood cleaner/brightener. Rinse the deck well and allow to dry for 2-3 days before staining.
The actual drying time depends on your humidity and weather. Add more time for humid days or if a dew forms in the mornings.