If I cast Leomund's Tiny Hut(LTH), the spell ends if I leave it. Spells and other magical effects can't pass through the barrier.
Can I cast spells from my familiar outside LTH?
If I switch my mind to my familiar inside the hut, and the familiar leaves LTH, does LTH end?
If I can switch to my familiar outside LTH, have I left the Hut, even though my body is still inside?
The duration of the find familiar spell is instantaneous
Many spells are instantaneous. The spell harms, heals, creates, or alters a creature or an object in a way that can’t be dispelled, because its m agic exists only for an instant. (phb 203)
meaning that after the initial casting there is no active spell to dispel.
Yes, while you are inside the LTH, you can cast spells through your familiar as the range of the spells is touch. The spell is cast by the caster with an origin at the familiar and has a range of touch. Never does a spell pass through the barrier. The spell does not pass through the barrier effect and is therefore not blocked.
These have been talked about in this question about what part of a familiar is magical
Having the familiar inside during the casting means that it won't exceed the creature limit and the familiar moving outside the hut won't remove the hut.
D&D counts bodies, not minds. You are still there unless you move out. Your familiar isn't you. Therefore, the hut will remain intact. (The DM may rule that your mind must be inside the dome in order to maintain the hut, but by RAW it doesn't.)
You’re missing one technical factor. The spell creates a hemisphere, therefore any spell that isn’t directed from you i.e. burning hands, but as per find familiar feature ..
... your familiar can deliver the spell as if it had cast the spell.
So the class feature treats the familiar as the caster .. ergo the spell isn’t going through the dome. It would depend on the specifics of each spell to see what could be cast or not, but I feel that this would work accordingly.
The relevant part of the spell's description is (PHB, pg 255):
Spells and other magical effects can't extend through the dome or be cast through it.
I'd say switching to your familiar's mind counts as a magical effect, and casting spells through it certainly counts (assuming the familiar is outside the dome and you are within). However, the familiar should still be able to pass through the dome and act, just not using your magical abilities.
To address your question "If I switch my mind to my familiar inside the hut, and the familiar leaves LTH, does LTH end?" directly, switching your mind up your familiar's would end as the familiar leaves the dome (since that is a magical effect), but LTH itself would not end unless you physically left it.
Again, for "If I can switch to my familiar outside LTH, have I left the Hut, even though my body is still inside?", you wouldn't be able to do this as you're trying to use a magical effect outside the dome.
