1. Many feel that Tywin wouldn't have supported Lord Frey in his plans for the "Red Wedding" if Jaime was still captive in Riverrun. Is that an accurate assessment?
"What if" is always a hard question to answer. Jaime's imprisonment might have made Lord Tywin more cautious, but there are no certainties.
2. Was "file Tyrek" closed in ASOS with the hint that he ended in a bowl of stew?
Maybe he did. Maybe he didn't.
3. That one is from me personally (it is a bit of burning issue for me because it debunked my favourite theory ;)), but - was Tywin really obligingly marching into Robb's trap when Edmure stopped him at the Red Fork? Did he really count on Lorch and Hoat with their IIRC 300 men holding Harrenhal and the northern prisoners in his absence?
Harrenhal is an immensely strong castle, and a garrison of three hundred is quite sizeable in medieval terms. Ser Amory =should= have been able to hold it. Lord Tywin likely thought that Roose Bolton might descend on the castle and besiege it, in which case Lorch could likely have held out for half a year or longer. The wild card here was Vargo Hoat changing sides.
4. Were circumstances and timing of Tywin's death something you planned for a long time or another case of characters "taking intiative", like with Cat?
That scene was largely written even before A CLASH OF KINGS was published. Hell, I'd been setting up that "Lord Tywin shits gold" line since his very first appearance in A GAME OF THRONES.