Since it is quite warm in Florida Zone 9b and will be for a long while, can I direct sow peppers and tomatoes in the garden instead of starting indoors? My plot gets about 6-7 hours of direct sun.
I would always start indoors sowing, even if I was in a warm climate like you are. One reason is that seeds and small plants outdoors have a chance they get eaten by animals or when they don't germinate you'll have empty spots in your garden. By sowing and germinating them indoors, you can use the best plants for the spots in your garden. So better chance of success.
For Florida zone 9, I found that you can harvest e.g., tomatoes twice a year. As you can see here, you are running a little late on schedule (they advice to sow peppers and tomatoes indoors end of July).
Yes you can.
IMHO only carrots MUST be sown in place... tomatos and peppers are optional, you can start them everywhere... just keep them mulched with straw and provide proper support with a trellis.