Yes, in terms of succession Melek holds the title of Baron (presumably of the nar Kiranka clan), so he'd be the logical choice to take over immediately following the death of the Emperor and Prince Thrakhath (in reverse order, technically).
As for the dreadnought, Melek's military rank is shintahr - equivalent to a Space Forces Colonel (O6). Assuming Thrakhath was the ships Captain, it's reasonable to believe that Melek would be next in line to command the ship (exactly the same situation as Bondarevsky taking over the Tarawa after Commodore O'Brien died).
Meleks official title is "chee'dyachee" (senior retainer) to Prince Thrakhath. After the war, his created and took the position of Chancellor of the Kilrathi Assembly of Clans.