Nope, I don't think so, anyways, from my past gameplay. I put them in my locker, and one day later, when I needed to trade with somebody, I gave them the Turnips. It happened for me, but I need somebody to confirm this.
true, you have to sell them by close of business on the Sat evening. (19th). As a new day starts at 6am, you have till 5.59am on the 20th, but you have to have somewhere to sell them.
You can store them anywhere - I tend to leave mine on the ground, untidy, but shrug.
The 7 day window you have to sell them, pertains to where you bought them - so you can sell them in another town set on a different date and time. In that sense you can take them back in time. (I've done it, have sev towns, set on different dates).
Any backward TT in your own town, and the turnips will rot.