I hope everyone enjoys reading these ideas! 
Oh I did! You got some decent brain juice going on, there.
Diamonds in a rough, if I may say so.
1) If the Shivans are dumping their heat into subspace are they also drawing their power from subspace?
Using capacitors and batteries for energy
only is insufficient because they are huge compared to actual reactors. That's why atomic submarines have a reactor on board and not giant car batteries.

Lets just say Shivans use zero point energy extracted from a subspace field within their reactors. No details, there, because "zero point energy" is a giant big problem on it's own. This also gives us an explanation why destroying the Lucifers reactors is so devastating.
Empty "space" usually does not absorb energy. "Space" in general are just dimensions or easy: "directions" you can travel along. "subspace" or "hyperspace" can be understood as nothing more but additional dimensions which are tiny compared to our normal space-time. You don't travel faster than light, you just use shortcuts, no laws broken.
Energy just travels through it. Using that explanation one can assume that dumping heat as radiation through a jump flash hole may be possible but you still have to radiate it away. I'm sure the Shivans found a way and if you ask them nicely, they might even tell you! ^^
The shield question, however, is a tough one but if you assume that shields have something to do with subspace it >can< be easy. First explanation would be that the Shivans advanced subspace technology allowed them to develop shields in the first place. How exactly they work is open to a lot of speculation, however. They don't work in subspace, because the physical principles which make them possible can not be applied in subspace.
Most sifi settings don't even try to explain shields, because any given answer to that question is just inconsistent and breaks so many laws that anyone would just say "what a giant pile of cr*p!"
2) Are they also recycling their energy back into the beams already to maintain the firerate they can pull off?
Exotic materials or rather elements:
Less likely. We discovered every presumable stable element in the periodic table, already. There are no other ways to combine electrons, protons and neutrons to form new ones.
Pardon! There are...
In theory you can just up the number of protons until you discover a new element but they are
all not stable and break down to simpler and stable elements within micro- and even just nanoseconds. Oh and they emit an unhealthy amount of particle radiation in the process. Exotic elements are rather unlikely and I could slap any sifi series which invents stuff like naquadah which magically turns into naquadriah because of "quantum events"? (Sorry, Stargate is not bad but come on people... do your research ^^)
Turning pure heat into useable energy violates the second law thermodynamics, which is a fundamental law of our universe. It's universal and the reason why we even exist. It's one of the reasons why the sun is shining. It's the reason why you can scramble an egg, but you can't "unscramble" it by simply turning your spoon in the different direction. The only way to achieve this is to reverse time and this is where believability goes in the pub to drink a pint because it won't deal with those explanations, anymore. ^^
To draw you a picture, here:
When you're on a mountain and you jump down you have a lot of potential energy which is turned into kinetic energy. You fall until you hit the ground. As soon as you did that all your potential energy is used up and you can not fall further. Of course, you could say "I will just dig a hole." But you need more energy to dig that hole than you get in potential energy. That's the same with heat. As soon as it's used or "pushed off the mountain" you're not able to use it anymore. If you want to do so, anyway, you need more energy than you'd get out of it.
I'd dump that idea entirely because it's very VERY unlikely that this would ever be possible. I'd rather put my chips on "extracting energy from space-time" and "zero point energy" because there is mathematical proof. However, it only works for a couple of nano watts.
(sorry... can't remember the actual crew compliment of an Orion).
I think it's 10000.
The bigger the ship, the lower the crew per cubic meter if you will so.
oh my, wall of text, again. I'm sorry ;-;