[RPG] Does a druid’s animal companion roll HD to determine health or just add 8 points for every HD

Probably.

There has been no clarification on the ambiguous "as if you had used Diplomacy" language. In general it's accepted that you can lay the Charm Person spell on someone, improve their attitude further with Diplomacy, and since those are two separate effects, when the spell fades they're left with however many levels of attitude improvement the Diplomacy did. As the Charm hex is of small benefit (1 level improvement) and short-lived (albeit a standard action and near undetectable), I see no reason not to allow it. Do note that the Charm hex works on animals and humanoids but Diplomacy only works on humanoids (whose language you can speak) so you'll need Handle Animal to get more out of animals.

Also if you're going this route please notice that before Helpful, there's additional checks required to get them to do things (with the Charm Person spell these are replaced with opposed CHA checks):

If a creature’s attitude toward you is at least indifferent, you can make requests of the creature. This is an additional Diplomacy check, using the creature’s current attitude to determine the base DC, with one of the following modifiers. Once a creature’s attitude has shifted to helpful, the creature gives in to most requests without a check, unless the request is against its nature or puts it in serious peril. Some requests automatically fail if the request goes against the creature’s values or its nature, subject to GM discretion.

It doesn't come pre-trained, but it does get the bonus tricks (obviously) and those can certainly include a General Purpose if you want them to and you have enough bonus tricks. Please note that a General Purpose is no different than the tricks that make it up except in terms of training time and effective check difficulty-- you get no benefit from choosing tricks that add up to a general purpose as bonus tricks. (There is a possible exception to this involving non-aggressive herbivores and combat training, but it's complicated).

Pushing an animal commands it to do a trick it isn't trained for but can perform, and is DC 25 (+2 if it's hurt). Because you are a ranger pushing your companion is a move action, v.s. the free action for handling it.