I'm not sure how I feel about the fact that it is a melee class though, since my paladin is also melee. Does a Blood DK play similarly to a Retribution Paladin to the extent that it will bore me?
Well, a Blood DK is a tank spec, while a ret pally is a DPS spec. So to that extent their focuses are quite different. In general, I consider DK's to be more interesting in playstyle than paladins - and no, I don't consider them all that similar.
In the end, I also care about their performance in PVP since I want my alt to be a PVP/arena character. How well does each class do in the Arena/RatedBGs? Is one more in-demand and considered to be better than another?
I have not heard much complaint from the DK's in the pvp front. They have a certain advantage in having multiple damage specs, improving the odds that at least one of them will be useful. There's no alternative DPS spec for priests, so it's shadow or nothing... and shadow has been commonly regarded as underpowered. That said, disc priests are highly favored as pvp healers, and if you enjoy healing, having two different styles to choose from can make the class more interesting.
It's also true that priests tend to take longer to level with, not only because of the 55 level head start a DK has, but also because their damage is primarily DOT, not burst damage which is more helpful in leveling.
The Mining profession would be more useful for you as a DK... for a few reasons. 1: your burst damage makes it easier for you to swoop down, kill any enemies near a node and mine it. 2: your pet makes it easier for you to mine a node while it distracts any mobs near it. 3: The bonus you get from the profession itself (stamina) is more useful for a DK.
All in all, based on your preferences and concerns, I'd say you're better off rolling a DK. The only reason you might choose a priest is if you just really want to play a caster, or you really want the ability to play a healer.