Agree with Thirith regarding #3.
Das dürfte vielleicht daran liegen, dass viele US-Polizisten auch im Militär gedient haben und oft auch noch in der Reserve dienen,
I wouldn't say that this has much to do with military service. They likely simply find it an efficient way of communicating (for various reasons in both branches).
Und wenn die Abkürzung regelmäßig schriftlich verwendet wird, gewöhnt man sich halt an, sie auch zu sprechen. Sprache ist nicht nur ökonomisch ...
That makes sense to me. Personally, regardless of the number of syllables, I find that it is slightly easier to say "GSW" than "gun shot wound" (perhaps because of the final consonants in the non-abbreviated words???). Also, unless you are trying to be extremely clear in your speech, many people contract "double-u" into something approaching "dubb-ya" or "dubb-you".
Edit: If "GSW" is used by "real-life" police, it wouldn't surprise me to find out that its origins are in written communication and that it has simply worked its way over to oral communication.