There used to be one, then it vanished and no one's bothered to give reason to its vanishing.

I... really don't think it's needed. Just bookmark it, or set it as your home page and check every 5 minutes.
Yeah I mean, we're already getting free stuff that we otherwise wouldn't have gotten, so just check it every now and then. Also sometimes if you visit after your 5 minutes are up, the shop may have cleared anyway so you could still have to wait.
I don't really think this is necessary.
I could see this being useful for mobile users, but I generally keep the free shop open in a separate tab and refresh every 5 minutes or so. It only opens so often and the wait doesn't seem to vary much (compared to pumpkin patch picking or dancing).
No. If you want free items, you have to make the effort to go and get them.
this is my favorite post EVER thank you Amber for remembering us that we are not entitled to free things