Since this is effectively the creation of a new Dracolich, the following are true:
- It would fail without an Adult or Ancient Corpse
- It retains all the features of the new body
Here's my rationalization of those two.
By inhabiting a corpse of a newly (or heck, not so newly) slain dragon, the Dracholich is not simply acquiring a new body (though it is that too), it is in essence becoming an entirely new Dracolich. Thus it would need a properly aged corpse as conversion is not possible on younger Dragons (it would also need to be a true Dragon, and not a poser).
The other half of this is that the new body is tied to a specific type of Dragon and thus has it's stats/abilities etc and not the Dracolich's previous. It's a new Dracolich and not a rerun.
If I was going to use this feature in a campaign though, I would feel very free to experiment beyond these bounds. An ancient zombie dragon that has found itself bound to a wyrmling's corpse for reasons unknown could provide a very compelling plot possibility.
The rules allow this, but...
There's nothing in the rules that prevents this from happening. However, there are a few things to consider about this tactic before employing it.
It can be immersion-breaking
Although it is technically within the rules of DnD to abuse the turn-based combat system for janky things like this, for some players this can break the sense of immersion that comes with good DnD. It can transform the fight from an epic tale about spell-slinging wizards to a videogame where you abuse the mechanics for maximum power. This kind of tactic is legal in the game, but does not make for a good story or in-universe character.
Any good DM can (and should) respond appropriately to shenanigans like this
For example, were I the DM here's how it'd play out. The first time you use Rope Trick as a sanctuary, some enemies might try to follow you up but it would mostly work out - this is what the spell is for, and it's a fun part of the game.
However, once you try the "climbing up and down" shenanigans, I'd just have enemies start readying actions. If they ready an attack, they can just attack you while you're on the rope. You could also have enemies stand at the bottom of the rope and attack your allies, and use attacks of opportunity whenever you attempt to climb back up. I also think it would be very reasonable to grant them advantage while attacking a climbing target. At this point, climbing up and down each turn is actually helping your enemies.
At the end of the day, there are a whole host of ways that climbing in and out is a poor tactical decision. That tactic leaves you spending a lot of time climbing up and down a rope, which is a poor use of movement and limits your positioning to a single location. While climbing you are fully exposed and can't use hands, but your enemies are free to reposition at will during their turns. Any slightly-intelligent enemy could use this highly advantageous situation to make your life miserable in a variety of ways.
You're missing the best use of Rope Trick
Rope trick is a fantastic spell for the use case you mentioned in the question:
[to] give a character in dire condition a brief respite against ranged attacks or spells.
However, it works better thematically and mechanically to stay in the sanctuary. You can take a few turns to use healing potions and spells, regroup, and reconsider tactics. You could even have your whole party take a short rest. Popping in and out every turn can be immersion-breaking and actually hurt more than it helps mechanically.
Best Answer
The monster will usually disturb the material it burrows through.
Consider the Earth Elemental:
The Earth Elemental (and the Xorn monster) can just glide through earth with its Burrow speed, without disturbing it. Monsters without such exception are assumed to disturb their environment.
Reading the Burrow rules:
Monsters like the Purple Worm and Umber Hulk have the ability to burrow through solid rock, and specifically leave a tunnel behind them:
Others, like the Ankheg and your exemplary Blue Dragon, also leave tunnels behind them. However, it depends on the environment whether the tunnels remain, or immediately collapse.
The dragon clearly has the capacity to create standing tunnels, but also to burrow out while leaving no chance of escape to unlucky adventurers.
Whether monsters leave a clear tunnel behind them, or the ground collapses behind them to block their path is environment- and DM-dependent. I would rule that, in lose sand, the hole would be covered. In hard ground, a tunnel would be left behind. In lose mud (like a mountain side), rocks would collapse and cover the tunnel again.