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.
It isn't defined
As you've noted, the spell description doesn't specify orientation. Windows are just openings and while most of think of a window on a wall, a window in a floor or ceiling is still a window and ropes can be pulled up into a vertical or horizontal window.
As user @MattVincent notes in a comment on the question, your spell description is not equivalent to the printed one. The changes are minor, but rope trick states (emphasis mine):
You touch a length of rope that is up to 60 feet long. One end of the rope then rises into the air until the whole rope hangs perpendicular to the ground. At the upper end of the rope, an invisible entrance opens to an extradimensional space that lasts until the spell ends.
The extradimensional space can be reached by climbing to the top of the rope. The space can hold as many as eight Medium or smaller creatures. The rope can be pulled into the space, making the rope disappear from view outside the space.
Attacks and spells can't cross through the entrance into or out of the extradimensional space, but those inside can see out of it as if through a 3-foot-by-5- foot window centered on the rope.
Anything inside the extradimensional space drops out when the spell ends.
It's up to the table/DM
If the orientation that you've been using has worked for the table, then you're good! If you want to keep it as such, you can. If you want to let your players pick an orientation at time of casting, you can.
And you can always change your mind if you think it's being exploited - but be sure to discuss it at the table so all understand how it works in the world you play in.
At my tables
We've imagined it in the same way that the image you link to does. A portal/window opens above parallel to the surface they're on with a rope dangling down.
Best Answer
This probably doesn't work…
The 2nd-level Sor/Wiz spell rope trick [conj] (Player's Handbook 273) likely can't be modified by the metamagic feat Reach Spell (Complete Divine 84) because the spell rope trick doesn't have the standard range entry Range: Touch entry that's needed for it to be considered an actual touch spell. That is, instead of the entry Range: Touch, the spell rope trick has the nonstandard range entry Target: One touched piece of rope from 5 ft. to 30 ft. long.
So it's likely that the 2nd-level Sor/Wiz spell rope trick [conj] (Player's Handbook 273) that's modified by the metamagic feats Reach Spell (Complete Divine 84) and Transdimensional Spell (Complete Arcane 84) (hence typically making the spell occupy or expend a 5th-level spell slot) just fails even if a caster outside the rope trick effect uses an effect to see inside the rope trick effect to aim the spell at a loose rope that a foolish creature therein left lying on the extradimensional space's virtual floor.
(This DM would've told the caster who was trying to prepare the modified spell that the Reach Spell feat's benefit couldn't be applied to the rope trick spell, and would've told a caster that casts his spells without preparation that he knew instinctively that the spell couldn't be modified with that feat's benefit. Likewise, this DM would allow any creature that was already familiar with the spell to make a Knowledge (arcana) skill check (DC 5 and maybe even 0) to realize the same thing.)
…And if it does, the DM says what—if anything—occurs
If the DM can be persuaded that a caster can apply the benefit of the metamagic feat Reach Spell to the spell rope trick despite the spell's nonstandard Range entry—maybe you brought beer—, and everything else is going that caster's way, that caster can cause
This would, of course, create in the extradimensional space an extradimensional space, which the rope trick spell warns readers is bad: "It is hazardous to create an extradimensional space within an existing extradimensional space or to take an extradimensional space into an existing one" (ibid.) Exactly how bad is the DM's call: the game never follows up this pregnant warning with rules except the specific case of the interaction between a bag of holding and a portable hole.
In fact, in the Rules of the Game Web column "Carrying Things (Part Three)" one of the game's co-designers, Skip Williams, says it's okay to disregard the dire warning accompanying the spell rope trick:
So, in the same way somehow tossing a bag of holding into a rope trick effect probably won't explode everybody hiding therein, zapping a loose rope in rope trick effect with a reach transdimensional rope trick spell likely shouldn't, for example, create a rift to the Astral Plane and cause the creatures within the rope trick effect to be forever lost.
Nonetheless, ask the DM. The DM may ignore the Rules of the Game columns (the columns do have a sort of infamous reputation despite sometimes offering otherwise unavailable clarifications), and then he can totally rule differently and make something spectacular happen… especially if you brought a lot of beer.