Adventure Class Ships Preview - The Extended Merchant

Do any of the Jovian Chronicles books have ship construction rules? And where are the rules for space combat?

I only have two of their books, the Spacer's Guide and the Space Equipment book.
The second edition main rulebook will give you the combat rules (Jovian Chronicles RPG 2nd Edition Player's Handbook). As for ship construction, you'll find them in the companion book (Jovian Chronicles Companion). Both are still being sold (at least in pdf format).