Each of Nidalee's cougar form abilities gets the bonus from Hunt, but only one of each. You get one enhanced Takedown, one enhanced Pounce, and one enhanced Swipe per Hunt.
I'm pretty new to AD Bruiser Nid too (I played a ton of AP Nid mid back when she was still powerful), but most pros I've seen go for Trinity Force first (I see Korean players go for Phage first, NA players go for Sheen first. Take from that what you will; Phage gives better mobility and AA/Takedown damage, Sheen gives greater burst during Cougar form and a little more mana and damage with spells to work with), then for a damage item (BotRK is common in NA, I've seen Korean players go for something like a BT, I'm currently experimenting with Essence Reaver for the mana regen. BotRK gives more damage per auto attacks, BT and ER give greater damage on her Q since she has a really great AD ratio on that attack; also enhances auto attacks, but they're more expensive, and don't give attack speed or the active that can save you if you're in a tight spot. Generally, Phage and Nidalee's innate mobility is usually enough, in my opinion, especially if you're good at orb-walking). After that, usually people go for tanky items, though I've seen a lot of people forgo a second damage item (ie. no BotRK, BT, etc.) and go straight for tanky items (Randuins, Spirit Visage, Banshee's Veil, Guardian Angel, etc. Resistances are a good idea for Nidalee over health, since she can heal herself, resistances make it so she can heal back HP easier than it is for people to take it away, if that makes sense.). It sort of depends how early you need your defensive items; some people who get really ahead go for a Last Whisper and Infinity Edge after their lifesteal/damage item. As for skill order, again, seen Korean players max Q first E second, like you would with AP Nidalee, NA players go for E first Q second. I guess it has to deal with your confidence in landing spears, also how much you want your heal to cost (it gets pretty mana intensive at higher levels, especially without any mana regen, which is one reason I'm experimenting with Essence Reaver). I personally max E first, since you're not going to be getting the massive spear damage from max range (especially since you're not building much more AP than a Sheen), also since you're usually within your AA range when you're fighting (lets you Pounce in when necessary) so the spear damage doesn't scale up.
As far as combos, it's pretty straight forward. You usually want to be in human form harassing with Autos when you can. You can use your traps like Caitlyn would, and choke off a lane, but watch your mana pool when you do this. Once you get a Hunt debuff applied, if you want to go in, you want to Pounce onto them, Swipe, and Takedown (W-E-Q really quickly). Your Takedown does more damage the lower health they are at, so Swiping first maximizes the damage you do. Using Swipe on the hunted target also lowers the cooldown of Pounce to 1 second, so you'll be ready to Pounce away nearly immediately. This is the pretty standard combo for short trades. Alternatively, if you intend to stay in the trade longer, you can Primal Surge (human form E) yourself before you jump in, giving you a heal and increased attack speed. You can save your Q for an execute later, but since you'll be getting a lower cooldown on Pounce, and since Takedown has a pretty low cooldown already, you'll be able to chase pretty well (you'll probably be able to get 2 Takedowns off if they stay in the trade for too long). If they get away from you and you can't chase, you can always Pounce towards them, switch to human form, and throw an auto or Javelin Toss at them (Pouncing in makes it easier to hit an auto attack, as well as a spear, but if you need more damage and are confident in your spear ability/don't think they can dodge, you can always Pounce away or for a better angle to throw a spear, since her spear does more damage if it travels farther than your auto attack range). Hope this helps
