Cavalier questions!

[Hogsy]

So I'll soon start a low-magic medieval campaign and I was thinking of what class to play. Me and my friends have usually rolled our characters using the 3d6+6 method for stats and we've been consistently getting very high stats(Starting out with 2 20s was very usual for me) and as a result we would get to higher levels of power much quicker. In this campaign we're thinking of using a 15 or 20 point buy system and I personally wanted to play a martial class with a synthesist dip for the flavour and for pounce. Since I'd generally get lower stats than usual the synthesist's physical stat feature would help me out a bit and if I stayed there for a couple of levels I'd get some evolution points to spend on an ability increase or two along with Pounce. I was thinking of what martial class to build and if it'd go well with the Synthesist so I have some questions for you. Generally my favourite classes are martials(Especially the Monk). Anyway, let's get to the point. I was either thinking of making a TWFer using the Ascetic Style chain and Pounce. It was either going to be a Fighter who gets the extra feats or a Monk who starts with IUS and I'd be able to TWF using FoB. I'm personally a bigger fan of Master of Many Styles due to the amount of styles you can use and how awesome they are thematically and it'd still work. But as I looked across the classes I noticed the Cavalier and I always wanted to play one. Do you guys have any tips for optimising a Cavalier? I've thought of a few basics like getting triple damage on my lance(By the way, I heard there's an item that makes your lance deal quadruple damage, does anyone have a link for that?) and what really made me consider playing the Cavalier was the Beast Rider archetype. I love huge beasts and apparently so does the Beast Rider! I could ride all sorts of crazy beasts and then later on get a Monstrous Cohort gryphon or something and even put some class levels in it! Anyhow, I wanted to ask whether I'd get a full-attack if my mount pounced. Secondly, I've read about the Cavalier that when you're mounted you choose to take the charge action along with your mount for the extra damage on your lance, otherwise you could just spend a standard action(if you're not using Mounted Skirmisher) doing something else or even be a little cheesy and use the Vital Strike feat. So I was wondering that if I did go along with the Synthesist dip and got Pounce would I be able to take the Pounce action when my mount Pounced to take a full-attack action with my Lance that would get triple damage on all hits? Mounted Combat is kind of confusing to me. Feel free to give me some tips and advice on dipping into Synthesist and playing a Cavalier. Tons of thanks in advance!