Yeah, I think you might be overthinking some of this stuff just a little bit...
There are probably two ways appearance changes can be handled:
1. Each form is separate. The two forms are set, but not directly related in terms of appearance changes. Meaning that altering the appearance "cosmetically" in one form doesn't affect the other in any way.
2. The forms are related. Major changes in appearance in one form will have a related effect in the appearance of the other form.
I would say that usually Shadowrun probably lands in camp 1, since the "tells" that one is a shifter already pretty well handles what a lot of people would expect for the appearance of forms being related. It would seem kind of weird (to me) if a wolf shifter doing something like shaving one side of their head would suddenly mean that in wolf form they don't have fur there, that's not really how animals work.
I usually include other features that tie the two forms together as well, like the classic "different color eyes" trope where both forms have the same eye colors. Or something like that, but casual cosmetic changes in one form doesn't really carry over.
As far continuing appearance of your metahuman form, there are a few more options there as well.
In general, the Shift power indicates that you have a set metahuman form, it is the same every time. So you are consistently the same person each time you shift into your metahuman self. Again, two options. Either A) Each time you shift, that form resets. or B) Your metahuman form has its own continuity. For example, your hair would be the same length as it was last time, and if you got it cut, it would still be that length the next time.
As far as dying goes, I would generally categorize that in the same way as cosmetics (makeup). It would be really strange to assume that makeup would stay put, and be even more silly to think that it would still be in place on the animal form after a shift. So I would probably rule that it would just go away.
Foci.
Yeah, a necklace is probably your best bet, but even still, you should be cautious about trying too hard to keep things between forms. Remember, clothes tend to shred when you change forms, even elastic has its limits.
As far as those qualities go, the cosmetic changes in their descriptions are mostly meant for Changelings, as they give the physical effects of that changeling effect. Since changelings don't have a standardized appearance, these visual clues from the qualities are used for what the SURGE did to them. The metavariant (and metasapient) races have pre-determined appearances, so you mostly take the effects from those qualities, not the physical markers.
Same thing goes for the Tail. Vulpine shifters have a tail in human form, but they don't get the quality. It is physically there, but it has no significant effect. The Tail qualities in Run Faster all have rule effects, Shifters don't get those effects.
Oh, I thought of another one...
It says Shapeshifters don't shed their animal hides when they shift. Does this mean Falconine keep their feathers?
I have no idea what you're trying to say here. Do falconine shifters have feathers in their metahuman form? No, just like other shifters don't have fur. You might be thinking of some depictions of werewolves where the wolf form essentially sheds its skin to reveal the human underneath, but that isn't how shifters work in Shadowrun. There isn't any fur (or feathers) left behind by the transformation, except potentially loose hairs (or feathers) that are sloughed off naturally...