When does the Free Shop close?

we users do not know. Sometimes it is a time limit, sometimes it is a total item limit. You can look in past years' news and see if there's a trend in the announcement maybe.

Would you think it was tomorrow?

my guess is that they're dropping a lot of Rare items, since most of the plushies and beanbags I have been getting haven't been in my collections. Subeta's team sometimes does this kind of thing. They will stop when prices drop to whatever goal they had in mind, so the "stop time" really depends on how players respond to the deflationary pressure..

Slightly irrelevant, but do you know when the "1st day of spring" that a lot items retire on is?

Usually the free shop goes until a few days into the new year cause sometimes they like adding Melody items for people who may have missed some days.

Also usually the first day of spring lands on March 21st.

yes, Subeta is biased to the northern hemisphere. The changeover may or may not shift a day or two if the site is having massive problems, but this is not very likely at all. You should expect it to be 20 March 2014 (according to Google).since the actual equinox occurs before midday.