Demoralize
There's various feats that add to demoralize, and the DC is usually fairly low. However, investing in it properly is a matter of considerable feats and even potentially class levels. A good standby in this situation (or any optimizing situation) is to look at what you'd be doing otherwise, compare the two, and decide which you like better.
I'd personally go with Demoralizing because as a melee character having as many mechanical things to do in combat as you can is always good - attack attack attack gets boring over 20 levels.
That said, after briefly looking at the options, Demoralize appears to be a genuinely useful thing to be doing, with the right feats and class options.
Using demoralize on the same creature only extends the duration; it does not create a stronger fear condition.
This is where I start to get interested. Not because being Shaken for longer is great, but because of this ability I found when I searched for 'is demoralize any good pathfinder'.
Frightening (Ex)
Whenever a thug successfully uses Intimidate to demoralize a creature, the duration of the shaken condition is increased by 1 round. In addition, if the target is shaken for 4 or more rounds, the thug can instead decide to make the target frightened for 1 round.
That's for the Thug rogue character archetype, and it receives it at first level. Let's take a look at what Frightened does for us in terms of mechanical effect.
Frightened: A frightened creature flees from the source of its fear as best it can. If unable to flee, it may fight. A frightened creature takes a –2 penalty on all attack rolls, saving throws, skill checks, and ability checks. A frightened creature can use special abilities, including spells, to flee; indeed, the creature must use such means if they are the only way to escape.
That is amazing. Takes someone completely out of the fight, and the best part is that penalties usually stack - so a Frightened, Shaken foe takes -4 to literally everything.
Better than that. The general rule is that if someone would make a Frightened character Shaken or Frightened, it instead becomes Panicked.
Becoming Even More Fearful: Fear effects are cumulative. A shaken character who is made shaken again becomes frightened, and a shaken character who is made frightened becomes panicked instead. A frightened character who is made shaken or frightened becomes panicked instead.
Panicked is another -2, and even if they can't flee, they don't fight, they cower.
Intimidate doesn't allow demoralize to stack with itself to create further levels of fear, but sources of Shaken, Frightened, and Panicked from other sources do stack with Demoralize (this is explained in the FAQ).
So if you can find a non-demoralize source of Shakening Baconing, you can stack that all the way up to 'they cower in a corner while you all wail on them'.
But that's broken!: Eh, not really. Note the 'Try Again' section in Intimidate. Every subsequent Intimidate check takes a -5, regardless of success. So you can't just spam this to victory - you need to succeed quickly on a few checks to frighten them for a round or two, they need to be vulnerable to fear, and anything too big or strong is likely to just resist your intimidate anyhow - it's still a good tactic, but for feat and level expenditure you're putting in it's not overpowered.
Now, this is already pretty alright, despite requiring a level in Rogue and skill points and intimidate boosting feats (maybe?) to pull off. But ideally you want to be applying this condition on round one, not after several rounds of increasingly ineffective intimidate checks. It would also be nice to be able to target more than one person.
Enter Enforcer.
Benefit: Whenever you deal nonlethal damage with a melee weapon, you can make an Intimidate check to demoralize your target as a free action. If you are successful, the target is shaken for a number of rounds equal to the damage dealt. If your attack was a critical hit, your target is frightened for 1 round with a successful Intimidate check, as well as being shaken for a number of rounds equal to the damage dealt.
With the Blade of Mercy trait (fluff it as hitting people with the flat of your blade, slapping them, kicking them etc - standard swashbuckler stuff), you can turn every attack into an intimidate attempt (yay). Combined with your high crit range (rapier), and TWF (off hand main gauche), and you could potentially fear multiple weak opponents or a single strong one in a single round.
Don't forget that using Frightening from Thug just makes the new intimidate check make them frightened - it doesn't 'upgrade' or 'remove' the Shaken condition, so...
Shatter Defenses, as it makes Shaken foes flatfooted to your attacks for this round and the next if you hit them when they're already Shaken. a) this is great for getting off sneak attack from that Thug level you took, b) this lowers their AC and also their CMD (trip them! kick them down stairs! swashbuckle! swashbucklers win the battle in the hearts and the minds before the first foe is struck!).
Shatter Defenses requires Dazzling Display, which is actually a good feat to have for this build regardless. It allows you to make an Intimidate check against everyone in 30' as a Full-Round Action, and you do it by showing off your sword skills! Talk about Swashbucklery. Normally that's a Waste of a Turn since Shaken isn't that great and if you're entirely surrounded by enough foes to make it worthwhile you should be tumbling to safety. But against weaker foes, with Frightening from Thug, you can show off a dazzling display of swordsmanship, Errol Flynn style, and watch an entire room of mooks lose their crap and flee in terror! Talk about awesome.
To optimize your Demoralize, I suggest you startize with Taunt. That way, instead of growling death threats, you can send them packing with witticisms about their mother! Or puns!
If you have backwards compatibility, the skill trick Never Outnumbered might come in handy, but you should steer away from Imperious Command, is it will result in flying DMGs.
Also, if you go this route, specialize your gear vs common fear-immune enemies like undead, and Paladins. Since that way you won't be a one-trick pony (Although this trick is cheap and leaves you room to develop your melee damage and whatnot).
It's worth noting that intelligent undead and the like actually seem to be affected by Demoralize - pathfinder uncoupled 'morale' from 'fear', so you can Shaken a Paladin or a Vampire, you just can't use Cause Fear on him.
No, you may not use reposition in an AoO in this manner, only trip, sunder, and disarm. Read the SRD rules on combat maneuvers, they each have a specified action type they can replace. For those three it is "an attack" but for reposition it is a standard action, http://www.d20pfsrd.com/gamemastering/combat#TOC-Reposition
Your flowing monk ability is completely separate, and procs on someone attacking you - then you can trip, or trip+reposition.
To get the AoO they'd have to provoke one normally, like by moving out of a threatened square, and to provoke the "flow" they would have to attack you - not an impossible combo especially if you have reach, but uncommon.
Reposition does not count against their move, but tripping certainly stops their movement.
The horse part is unclear and probably a separate question anyway.
Best Answer
I would argue that Tim's answer is not accurate. I understand that his proposed solution is not based on raw, but it is based on the idea that "your opponent must fight gravity to advance".
That it is just not the case for this rule. Higher ground advantage does not seem to emanate from a difficulty your opponent has, this is already covered on a separated rule for fighting uphill. It seems to come from an ease to strike from having gravity aid you. Swinging down is easier than swinging up.
This is further evidenced by the rule on Combat while Mounted that specifically determines you get high ground bonus while mounted if your target is smaller than your mount.
It is still up to the GM to call what high ground is, I would recommend simply giving higher ground advantage to any creature who is at least 5 foot higher than another.