If you have an animal Companion that has poison, such as a Fleshraker Dinosaur, is it possible to "milk" these (As one would with a poisonous snake) to obtain their poison, and apply said poison to one's weapons? If so, how much poison could one obtain and how much time (and possibly other resources) would this take?
Yes, there are rules for using Handle Animal to milk animals for poison in Drow of the Underdark. Well, technically, the book has rules for getting venom from vermin, which requires the Vermin Trainer feat, but most everyone allows those rules to be used for regular animals as well, and without the feat.
Anyway, you have to teach the animal the Bestow Venom trick, and then it is a DC 15 check each time you want some poison out of them. This is much easier and cheaper than crafting poison.
For the record, milking venom from animals does no harm to them, in D&D or real life. It usually just involves getting them to bite or sting some cloth over a jar, so the venom injected goes into the jar. (Humans milk venom because in real life you need it to make the corresponding antivenin—in D&D alchemical antivenom just works on all possible poisons and doesn’t seem to need any poison as an ingredient, but that is fantasy.) Regardless, the rules do not include any damage to the animal, and suggest no alignment for the act. Using venom, or any other poison, remains generally illegal and specifically against a paladin’s code of conduct as too dishonorable, implying the use of poison is somewhat chaotic, at least when in violation of laws against doing so.