If they were stuck in a time loop, how did Hatter get out to get the White Queen out of Witzend on Horvendush day and how were they able to interact with Alice when she visited them as a child?
Time said they were stuck in a time loop until Alice arrived. Which she eventually did as a child during her first trip to Underland. Once she showed up for the tea party as a little girl, the time loop was broken.
There's a problem with that though. In "Alice in Wonderland" when Alice arrives at the tea party for the second time, Tarrant says that he "killed Time" while waiting for her return and "Time became quite offended and stopped altogether; not a tick ever since." A moment or two after this he pulls out his pocket watch, looks at it expectantly, and says, "I'm waiting." Thackery then pulls a pocket watch out of a tea pot and says that it's "ticking again." The Hatter then grins at his own watch before putting it away. The audience is left to assume that his watch has started ticking again, too. So that means the time loop wasn't broken until Alice's second trip, and therefore, under the rules of the loop that we're shown in "Alice Through the Looking Glass", Tarrant couldn't have been serving under the White Queen, been present at Horevendush Day, or have taken Mirana to safety when the Jabberwocky attacked. This also means that Mally wouldn't have been able to be among the group that greeted Alice when she arrived the second time. (Like I said in the Continuity Snarl entry, whoever was supposed to be checking the continuity should have been fired.)
Or, if we want to get into some WMG, perhaps Tarrant tried to do something similar to Alice's time traveling at some point after Horevendush Day. We know he went a bit mad(der) after the Red Queen attacked with the Jabberwocky. He might have tried to find a way to go back in time to prevent it. And if we have a bit of a Stable Time Loop, he would know it is possible to go back in time because he knows Alice did/will do it (though not necessary the dangers of accidentally running into his past self). And while Thackery was too bonkers to realize this was a dangerous idea, maybe Mally tried to find someone to talk some sense into the Hatter. Needless to say, the plan doesn't go quite as intended and Time stops him. Maybe Tarrant or Thackery asks if Time intends to "trap them at one minute to tea time again until Alice shows up." And while Time is a little confused (because that would be the action of future Time rather than the current one), he shrugs and goes for it. Yes, this is really stretching things in order to fill in some plot holes, but it is at least theoretically possible.
In the supplementary book A Matter of Time, the first point is confirmed as true - the SECOND time Tarrant was stopped, that can presumably be just after Horunvendush day, perhaps on Shatterky / Fustilug, where understandably Tarrant would attempt to kill Time, making him stop the trio. Easy as that.
Whatever happened to the subplot about Alice being put away in a mental institution? She returns from Underland once and wakes up there, then escapes, goes back through the mirror, then returns later and finds...her mother just finalizing the deal with Hamish, and it's never brought up again.
I assumed that was a temporary example of the Timey-Wimey Ball, as Alice witnessed what would happen when she nearly destroyed history in Underland; most likely the asylum was where she would have been sent if she'd never been able to travel to Underland and experience the Character Development of the first movie, encouraging her to face her responsibilities and take more risks. When Alice returned to Underland and completed her efforts to save the Hightopps, it erased that time in the asylum and 'reset' history back to the chain of events where she went to Underland after her first meeting with Hamish and then returned with no time having passed.
What's going to happen to Iracabeth after the events of the film? Sure, Mirana apologized for eating the tarts when they were younger, leading to their reconciliation, but the reasoning for her crimes still didn't justify them - she beheaded a lot of people and subjected Underland to years of tyrannical rule, all of which Mirana imposed a punishment for. How is everyone going to react when their queen brings her murderous sister back to live with her at the castle?
Mirana might justify taking her sister back in as "rehabilitation", to help her be a better person and no longer a threat to society.
Why couldn't Alice just have gone back to the point where Hatter found the hat he made as a child and prevented him from discovering it so that he doesn't have the depressing revelation about his family?
Two reasons: The Doylist one being that if she had, there wouldn't be much of a story, and the Watsonian one being that doing such a thing would be Cruel to Be Kind, at best, which is a great deal more cynical than Alice, Mirana, or anybody involved in the plan would consider.
So Alice is rushing to return the Chronosphere, but time freezes before she is able to. But then, after a few seconds, it works somehow, and everything is normal. What? note I'd like to remind you that time is frozen, so "seconds" should not matter.
As was shown in the film, Alice had managed to bring the Chronosphere close enough to its pedestal before time stopped for a spark to be able to "jump" between them and connect them, which was enough to restore the clock to its functional operation.
During the rust plague, it looks as though the rust has to touch you to convert you. Could the Cheshire Cat or other flying Underland residents survive that way?