No the bag will not burst, because Rope Trick is not an item
The key word in the caveat you cite is "item":
Placing a bag of holding inside an extradimensional space created by a Handy Haversack, Portable Hole, or similar item instantly destroys both items and opens a gate to the Astral Plane.
The two examples the text gives are also items.
However, the extradimensional space created by Rope Trick is created by a spell, not an item, and so does not fall under that clause.
There are many spells that generate extradimensional spaces (such as Mordenkainen's Magnificient Mansion)--if the designers intended the Bag of Holding to explode when it entered such spaces, they could have easily made the text refer to any extradimensional space, or used spells as examples.
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.
Sure, why not?
I see no reason this wouldn't work; ie, you're hovering there however it is that you're hovering, and you cast rope trick. The rope goes up from where you cast it.
And there is nothing in the spell description that says this wouldn't work. You mentioned that maybe the phrase "rises into the air" means it has to be on the ground, but that is not the plain English reading of the phrase "rises into the air". A balloon, floating skyward, at any point can be said to be rising into the air. Rising into the air is not restricted to something on the ground.
Or to put it another way, "you can certainly try."
Ask the GM
Really, the best person to ask is the GM. This is one of those circumstances where the rules don't really seem to have anticipated the unique use you want to put the spell to. That's great! That's one of the things that makes D&D so much fun. But really, you need to go back to first principles: describe to the GM what you want to do, and the GM will narrate the consequences of your actions.
If, by any chance, the GM is unsure of whether it should work, there's nothing in the rules that says not. If through some unanticipated quirk allowing it to work causes some game-balance issue, the GM can rule differently in the future. It's perfectly okay for the GM to say, yes, I allowed this before, but it's broken and we need to reign it in, providing an in-game explanation if needed. In this case, it's hard to see what that quirk would be.