Some random observations:

Yes, if you want to transpose a key up or down a semitone, you'll either be adding or subtracting 7 flats or sharps, which causes the key signature to shift from flats to sharps or vice versa (or to C), and the sum of the flats or sharps will always be 7. This is basic music theory.

Furthermore, there will always be one semitone-adjacent key which has the same staff letter and a legitimate key signature (no more than 7 sharps or flats), so you can transpose to that key "as written", except for translating accidentals. C is unique in being transposable to two adjacent keys without changing the root letter--C# (7 sharps) and Cb (7 flats)--, but who wants to use either? Otherwise, your choice is pretty much forced. For instance, D can be visually transposed to Db (5 flats), but not to D# (9 sharps--too many!)

The same trick applies to the minor keys and the other modes (eg., Dorian), though some pairings change--D minor (1 flat) must be paired with D#m (6 sharps) rather than Dbm (8 flats); compare with the example above. So it's not quite as easy as learning the pairings for major and using the same pairings for other modes.

Regarding key choices:
Capos are designed to work in conjunction with frets and flat keyboards; a violin has neither (though I'm sure some clever instrument-builders sell special capos which act as moveable bridges.)

I wouldn't say classical violinists eschew open strings, but rather that the nature of the music usually precludes using open strings. Vibrato is used extensively on sustained notes, and is impossible on open strings. An open note cannot easily be ornamented, except by grace notes or trills from above. Classical music uses far more modulation, parallel voicing and accidentals than in most fiddling.

On the other hand, as previous posters have mentioned, open strings come in mighty useful in fiddling, which makes extensive use of drone strings and the natural harmonics of the standard tuning. An open string also simplifies making position changes and ornaments on the next higher note (as on guitar). So for fiddlers, the use of open strings has more influence on key choices.

A number of instruments are quite limited in the keys--or should I say key signatures--they can play. A tinwhistle only really plays two keys easily. You can get them in most of the standard keys, but the ones for some keys are just too shrill, breath-robbing or clubby for serious use.

A hammered dulcimer is limited to the 3 or 4 keys for which it was designed. The keys always progress by 4ths, and only a handful of accidentals are available in each key (a byproduct of the diatonic notes in the related keys.)

An Anglo concertina is limited in the chords it can play in each direction--you'll often find an Anglo concertinist singing only in the key of C or G. (Where's my pistol?)

The English concertina is fully chromatic in both directions, but playing accidentals (on the outside rows) can be a bit of a trick, particularly when you have to chord two buttons with the same finger. If you transpose to a new key, you have to use an entirely different finger pattern, particularly if you go up or down a 2nd or 4th, when you have to switch hands/sides. Besides which, the generally available concertinas aren't quite as well-tempered as they ought to be--downright ill-tempered in some keys.

When I sing, I often use a much wider range than the melody itself spans. I find that if I shift the key by even a semitone, the song may become noticeably harder to sing. Or it may lose some of the deepness or lightness I wanted. When I can, I put a song into an "unusual" key, just so there is that distinction and mental shift from one song to the next. I wonder how many songs get written down in strange keys just because of personal peccadillos like this.