Set film lower than rating?

Some films are very good at this. Tri-X at 200 is a good one, and Foma 200 at 100. Other, not as much. It's usually not really necessary. Look at dev time charts because you might need to adjust the time to avoid overexposure.

In negative film, I can't see that it does much but make a harder-to-print negative. Maybe that's just me. There was a day of high variability in films, cameras, and processing that the manufacturers rated films lower than they later did, and it might be some sort of ancestral memory at work here....

For color slides, on the other hand, many people like the darker, more saturated colors that come with slight under exposure, so many people shot ISO 25 Kodachrome at ISO (actually ASA) 32 in the days back when.

Depending upon the lighting you can gain additional shadow detail. If lighting is flat you might not want to downrate it. For example, I usually expose Plus-X at box speed, but under flat lighting or overcast days I rate it at 200.

Over exposed negatives look good as you hang them up to dry, but a correctly exposed negative prints best. So no, use the box speed IMO.

there was a Modern Photography article about taking photos at or before sunrise or omn misty days and rating asa 25 at 32.
it incerased saturation when there was flat lighting few shadows and lower visual contrast
with negative B&W film there was a 2x safety factor. in the 1950's.
most B&W films were rated at half speed.
tri-x at 200 not 400.
many advanced amateurs and pros rated the film at 400.
when kodak rated the film at 400, there was no change in the film.
just the rating.
slide film can only be FUDGED a little when the lighting is flat.
the newer color negative films are quite flexible.
back then there were only kodacolor films in roll film sizes, no 35mm!
and one of the instructions ( then) was to use "direct sunlight"

Yeah, and wasn't ASA 12 considered fast for a color film? :)

It's true, current color negative films have remarkable exposure latitude. Somebody on the forum posted a test a while back where he overexposed Ektar by 0, 1, 2, 3, 4 stops... it took a while before you started seeing real problems.

When fast color negative films began to improve, many users discovered that exposing at a lower E.I. gave finer grain and better color saturation. I'm sure that the lab really made a difference here. I usually rate Portra 160 VC at E.I. 125 for weddings (E.I. if church has dark walls). YMMV, though as some labs that receive down-rated film will deliver washed out prints or scans.

Not always. It depends on the light and thus the subject brightness range. On a bright day I typically downrate film by 1 1/3 stops (e.g. HP5 at 160 ASA or FP4 at 50 ASA) and then cut development by 45 percent. It will give you a negative with a longer scale of tones.

If you search for 'finding your personal film speed' you will find lots of articles about doing tests at varying exposures and developing for various times until you find a combination that suits your methods.

I didn't bother with doing the tests myself as I noiced that nine times out of ten, the results were to halve the ISO and reduce the development by 20-25%

I tried that with HP5+ and liked the results so continued doing it.