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Here's a sample that I already posted re: fingering.
The last four chords could be played with 1-3 slipping down from F#/A# to G/B and then hitting the A/C with fingers 2-4, etc.
Would it ever make sense, or is it ever proper form to just slip down from the black keys to hit the next pair with the same fingers to save some complexity?
The short answer is "yes".
A slightly longer answer is that sometimes there is no better option. It's not ideal, though, because it doesn't give you a lot of control over what you're playing.
In this specific case it's hard to tell since you didn't include the next bar. The standard fingering for four climbing thirds would be 1-2 1-3 2-4 3-5, but this is also problematic here. Depending on what comes next (not included in the picture of the sheet music), you might not want to have a five on the last note, since I assume the line continues upwards, you would have a 5 on the C#, a black key and I think this fingering would feel rather cramped (but probably your hands are different anyway). Depending on the next bar I would probably play this with 1-2 1-3 1-4 2-3, ending on 2-3 so you could easily continue upwards (hopefully my guess about the next bar is correct) and you don't have a 5 on the c# anymore (you would still have a thumb on a black key, but I don't see any viable way to avoid it).
Edit: I didn't consider the preceding notes, since you already have 1-3 as a starting point. Seeing this d and e, though, I would suggest 2 1 2-3 1-4 1-5 2-4.
It's a very common technique to do that on one note of a chord with your thumb. Doing it on several notes might lose control of the dynamics and/or the rhythm, especially in a fast tempo, because there is nothing that is "anchoring" your hand position while you are doing it.
In the specific example, 1-2 1-3 2-4 might be better than 1-3 1-3 2-4. But putting your thumb on the F# doesn't seem a good idea IMO, unless you can't find an alternative.
Yes. Do whatever achieves you goal, while minimizing harm, as in every other aspect of life.
Arguably, "sliding" a finger from a black key onto a white key makes it more difficult to achieve legato. So when there are other options, it is not usually adviced. But sometimes legato is not a required virtue, or the situation is too difficult or impossible to "cross one finger underneath another".
This mostly happens when you have no time to "cross the finger" because the tempo is fast, or disallowed to do so for other fingers are pressing another note(s). And most frequently this technique applies to the thumb, because one's thumb (in either hand) is effortless to be "placed onto" something with precision. (Difficult to express this verbally, as I am not English native XDDD )
You can see either is true in the following examples. It took me some time to find these examples. It is most persuasive to cite those passage when a score-press editor suggests so. (I believe these are all Henle Urtext edition, but I am not sure, as they are copies (shh...!!).
The tempo of this section of Chopin Ballade 3 is fast, and pedal is used, allowing possibility of sliding l.h. thumb.
The tempo of the finale of Beethoven op.101 is fast, and in either case, 5 is used and it is barely possible to "stretch his hand" and cross finger. The 2nd red arrow is closer to the spirit of your cited example.
The tempo of the finale of Beethoven op.101 is very fast, and the insane requirement of r.h. to play another voice makes it only solution to slide r.h. thumb. The 2nd and 4th red arrows are closer to the spirit of your cited example. This is considered one of the most difficult piano pieces of all time, so be warned if you want to try this. (I have not.)
