Assume I'm flanking someone and I Sneak Attack them. The next turn, I'm still flanking. The enemy is still sandwiched between me and my ally. Can I keep Sneak Attacking?
Yes, despite the name, flanking lets you Sneak Attack without actually being sneaky.
When flanking someone with the help of the an ally, your rogue has no need to be stealthy in order to benefit from the Sneak Attack ability. The ability's rules just let you do it.
The idea is that, when someone is flanked, they can't defend as effectively — either way they turn, they have to put a side or a back to an enemy. The Sneak Attack ability, despite the name, really means the ability to exploit someone's weakened defenses to attack them in a vulnerable spot (PHB, p. 50, emphasis mine):
If a rogue can catch an opponent when he is unable to defend himself effectively from her attack, she can strike a vital spot for extra damage.
Sneak Attack does also let you hurt people more when you attack them stealthily — but stealth gives that opportunity for the same reason flanking does: because they're unable to properly defend against the attack. That's what it always comes back to: do they have vulnerable spots they're not defending properly. Flanking someone creates that opportunity.
Yes, usually that rogue can continue to deal sneak attack damage
The rogue's special ability sneak attack says
If a rogue can catch an opponent when he is unable to defend himself effectively from her attack, she can strike a vital spot for extra damage. Basically, the rogue’s attack deals extra damage any time her target would be denied a Dexterity bonus to AC (whether the target actually has a Dexterity bonus or not), or when the rogue flanks her target. This extra damage is 1d6 at 1st level, and it increases by 1d6 every two rogue levels thereafter. (Player's Handbook 50)
(Emphasis mine.) There's no limit on the number of times a typical rogue can deal the rogue's sneak attack damage to a typical flanked creature. The foe's the meat in a sword sandwich.
To be clear, this reader finds it cruel of the Player's Handbook to follow the quotation above with sentences like, "Should the rogue score a critical hit with a sneak attack, this extra damage is not multiplied," and, "Ranged attacks can count as sneak attacks only if the target is within 30 feet," and, "With a sap (blackjack) or an unarmed strike, a rogue can make a sneak attack that deals nonlethal damage instead of lethal damage. She cannot use a weapon that deals lethal damage to deal nonlethal damage in a sneak attack, not even with the usual –4 penalty" (ibid.) (emphases mine). If one's trying to wrap one's head around the rules the first time, to use jargon like that right after the description of what sneak attack really means is misleading—in those followup sentences the game makes it sound like a sneak attack is something special that the rogue does rather than just extra damage that's usually simply the product of battlefield conditions and positioning.
That is, as the opening sentences indicate, a sneak attack isn't actually any kind of real sneak attack as the term is used by normal humans. There's nothing particularly sneaky about the attack—the rogue doesn't, for example, like, lose the element of surprise once the rogue's made a sneak attack (surprise being a whole 'nothing thing), nor must the rogue that's flanking a foe necessarily break line of sight from the foe to make another sneak attack against that foe or anything like that. Instead, the term sneak attack is used in those later sentences as shorthand for an attack that deals the rogue's normal damage plus the rogue's sneak attack damage.
Thus the term sneak attack is actually closer to sneak attack damage, and the rogue deals that extra damage—that sneak attack damage—when the rogue successfully attacks a foe that's vulnerable to sneak attack damage that…
- can be flanked when the rogue is one of the creatures doing the flanking (e.g. a barbarian of sufficient level might be immune to being flanked due to the barbarian's special ability improved uncanny dodge, a creature like a beholder may have the special ability all-around vision that prevents it from being flanked) or
- is denied its Dexterity bonus to Armor Class (e.g. the foe is flat-footed, the foe is helpless).
A handful of other situations also allow the rogue to deal sneak attack damage (e.g. the 2nd-level Sor/Wiz spell wracking touch [necro] (Spell Compendium 243)), but these individual exceptions are called out by the exceptions' descriptions.