What are the purposes of the various specialty knives?
The standard answer is that you actually need a grand total of three knives, and could conceivably make do with one.  The usual load-out recommended is 1)6"-8" chef's knife, santoku, or cleaver, 2) 3" paring knife, and 3) long serrated knife.  The chef's knife or santoku is for most work (a cleaver can be used for this if you're a mad man), the paring knife is for small work (fine vegetables and mushrooms), and the serrated blade is for crusty breads and tomatoes (which can end up crushed by straight blades).   If you are embarrassed for funds, ditch the serrated knife and paring knife, in that order.  If you're doing serious butchery, a 10"-12" carving knife and a small thin jointing knife are worth the investment.    There's a special knife for fileting fish, but to be frank I've had one for 20 years and it's never been the best tool for the job.

Watermelon can be sliced with a large straight blade (I might pull out the 12" for that), but only with a lot of gumption.  Show fear or hesitation, and you're going to make a mess. 

More important than what you buy is how you use it.  Search on youtube for "introductory knife skills", or take a cooking class on it.  Well worth the investment. 

(As should be obvious from the above, whatever you do don't buy a "knife set".  You'll end up paying a lot of money for knives you will probably never need.)   

Except for raw tomatoes, french bread, and peeling fruit, everything on your list ends done by my 8" santoku (or 6" ceramic, which is not something for beginners, purely on price).