Core Bestiary

They can cross over (by flight, bridge, or being carried) running water, but not pass THROUGH the water (swimming, etc) lest they risk permanent destruction, as their mist form escape doesn't function when they are slain by running water.

Pathfinder Splats

Classic Horrors Revisited very briefly mentions them being thwarted by running water, but doesn't specify they can't cross it, so I'm assuming that's a reference to the damage they take if immersed. It also very specifically says that many vampires don't adhere to the generic vampire rules, as I point out below, and relying on a single "traditional" weakness could be an adventurer's last mistake.

At the table

While your question asks for RAW, with all our talk of vampires (now spanning three questions), I feel I should mention that vampires are one of the most, if not the most, house-ruled monsters there are. Adding and subtracting classic weaknesses, separating them into clans (along lines such as games like TES4:Oblivion or Vampire: the Masquerade), or simply giving them class levels you might not expect (It's a vampire... cleric of a neutral deity? Channeling negative, oh dear...); all of these and more are ways vampires quickly find themselves telling RAW to bite them (pardon the terrible pun). If your vampire friend finds himself stymied by running water, great; beware your DM didn't remove a different classic weakness to add that one. If the vampire is in full-plate, don't assume that means he can't cast fireball; the Tremere clan's blood magic cares not for your arcane-spell-failure rules.

Yes, but it's very unlikely.

Trying to hit a specific location on a creature requires the variant rules for called shots (Ultimate Combat p.193). Which even specify how they interact with vampires and cyclopses.

A called shot is an attack aimed at a particular part of the body, in the hope of gaining some extra effect from the attack. The smaller or better guarded the area, the more difficult the called shot. A called shot is a single attack made as a full-round action, and thus can’t be combined with a charge, feats like Vital Strike, or multiple attacks with a full-attack action.

Some effects in the game, like true strike or the flash of insight ability of cyclopes, provide automatic or nearly automatic hits. Using such an ability on a called shot turns it into a normal attack, with none of the benefits or penalties associated with called shots. From a story perspective, this is because the effect cannot distinguish between a hit in general and a hit in a particular area, but it’s also necessary to keep the power of such abilities in line with their original intended effects. Some Game Masters may prefer a more theatrical or dangerous game in which magic can make a shot through the eye nearly certain, in which case this rule can be ignored.

The consequences of a successful called shot vary depending on whether the hit is a normal hit, a critical hit, or a debilitating blow (a hit that deals half the creature’s hit points of damage (minimum 50) or more).

Heart

Called shots to the heart are challenging (–10 penalty).

Called Shot: A called shot to the heart is just a normal hit with no extra effect.

Critical Called Shot: [...] A critical hit to the heart against a vampire made with a piercing weapon composed entirely of wood leaves the vampire impaled through the heart by the weapon if it fails its Fortitude save, with effects as described in the vampire monster details.

Debilitating Blow: [...] A debilitating blow to the heart against a vampire made with a piercing weapon made entirely of wood affects the vampire as a critical hit to the heart; for example, it still receives a Fortitude save to avoid the consequences.

So it's theoretically possible, but you have to pass all of the following criteria:

• Perform a single attack as a full-round action. (Unless you've taken a feat to relax this constraint.)
• Hit while taking -10 to the attack roll, and without using Flash of Insight or anything else that would trivialize this step.
• Either get a critical hit or deal at least 50 damage or half the vampire's hp (whichever is higher).
• And finally the vampire gets a fort save. (DC = the numeric result of your attack roll, i.e. the highest AC it could have hit).