Anyway, the questions:
Are insects and bugs in general included in animal rights?
Yes, if and when they decide to grant themselves such rights.
What does this mean for parasitic animals such as tics, heartworms, headlice and fleas?
Only if the animals to which they are parasitic do not grant themselves rights to NOT ne used and have better defense of their right.
Is it better to kill them, or let their host suffer?
Why compromise? Do both. Let the host suffer and them kill them ... <just kidding>
Also for exotic species in our care that eat crickets, is it wrong to through them into an enclosure to their certain death? (assuming the animal eating the crickets can not be released back into the wild for whatever reason)
No, as it is a choice and as long as one is comfortable with the choice and will accept the results of the choice.
What about obligate carnivores in our care?
In our care means we are obligated to feed them their proper food.
Is it wrong to raise and breed rats for the sole purpose of killing to feed to snakes?
No more so than raising sunflowers to feed the seed to birds.
Again, assume this is a rescue snake that cannot be released into the wild. The same goes for cats, ferrets, etc. (This isn't a question about the ethics of keeping pets, more feeding and caring for animals that are unable to get themselves food for whatever reason)
I see no problem with this.
On a similar subject, what about wildlife rehabilitation centres? Would the answer to this be we should not interfere with nature and let them die in the wild rather than trying to help them?
Given many of the rescue situations are the result of human action or due to human creations, it is our responsibility to provide such care.
Should we be caring for all animals equally, even if they are not companion animals and living wild?
We cannot care for animals equally because animals are not equal. Those which are prey cannot be saved from the predator without dooming the predator and inversely the predator cannot be saved without dooming some of the prey animals.
On a slightly different note, is it exploiting a dog if you teach them tricks?
No, unless there is strong negative actions used to force the trick.
What about if you preform these tricks in front of people for money?
Working dogs earn their room and board.
And the dog is not forced to preform, but sees it only as a bonding/playful experience with a human?
Dogs also like to teach tricks to humans. If they look a certain way we humans will take them outside.
Now, I'm sure this is probably more to do with personal opinion, but what is your stance on making things with parts of dead animals?
As long as the animal is dead, I see no problem with it.
Such as taxidermy out of a roadkill fox, or collecting bones from the forest, etc. I guess it would be interesting to hear your take on the use of preserved human remains too, such as what Gunther Von Hagens sells. Also, what of animals that are already dead, what should become of their remains? (Antique fox fur stoles from the 1940s, taxidermy mounts in museums, etc)
No problem. I actually came close to purchasing some old museum exhibits from the turn of the century a few weeks back. My wife was not opposed to it but the display area just was not in our house for either of them due to the size. (Native squirrel and pheasant species)
My final question is something I feel quite strongly about - Vegan Dogs. (I'm aware cats can go vegan too, but I'm so very against this for many reasons it's probably not something to touch on here! )
Dogs can live on such a diet, but they will go crazy for some meat too. I believe treats are due them since it is their natural diet too.