1.) You can't target the 10' space directly under the hut by spellcasting rules. You don't have line of effect (what with the dome being opaque and not letting spells pass through) unless the hut is on a pedestal.
2.) The spell takes a minute to cast. Even if you could cast there, the spellcaster would be plugged full of arrows before they finished casting.
3.) The dome still wouldn't fail. At most, the characters would just drop through the hut. But that's a somewhat dodgy interpretation, since the Earth Elemental wouldn't displace earth. How are you moving through the square if the portion that isn't blocked by Earth Elemental is filled with non-displaced earth? The Earth Elemental can't even create a Bugs Bunny-style hole.
But it can't enter the hut.
You are missing the point. However I'll reply anyway.
The person who had a tiny hut attached to a glyph of warding that was inscribed on a charm that could be smashed on the ground to activate tiny hut was breaking the rules far more directly than my counter example that had the conjure elemental to offset it. You may not have seen how I got involved in the thread, but read up and you'll note that glyphs of warding can't be used that way.
The spell states that you are targeting a 10' cube of earth that's "in range of the spell". I'm aware that LOS considerations and other things come into play with spells but one interpretation of the specifics of the spell could very well be that you don't need to have LOS on the 10' cube of earth under the hut because you have LOS on everything around it and it's in range of the spell. Certainly that's a far more agreeable argument than breaking both the "you can't move the glyph 10' from where it's transcribed in the first place OR even worse, putting a beneficial spell into a glyph to begin with.
Regardless, just cast it on the 10' cube directly outside the cube. First round the elemental uses earth glide to move down, next round he moves over, last round he moves up. At most it's three rounds to kill the hut. Earth glide doesn't displace earth so it's a function of just going through the motions.
Now to your direct points.
1. See above.
2. The caster would be riddled full of holes if the party knew he was casting - agreed.
3. According to the spell description for Tiny Hut the spell fails if a large sized creature is in the area. I don't judge that to mean at time of casting only.
Be well
KB