You have a Harlequin Type Phalaenopsis Hybrid. Harlequin because the spots and blotches vary from Flower to flower and year to year.
American Orchid Society Culture notes are in the link.
Plant is in reasonable condition. Do not be in a hurry to do anything. Do some research. Cut off the ends of those obviously dead roots. Make sure there are drainage holes at the base of the pot.
Come back here if you have more specific questions, do not forget to tell the posters where you are in the USA.

Here is a link that might be useful: AOS Phalaenopsis Notes

Thanks for the advice, and the link! And sorry, I didn't even think of mentioning location since I have it indoors, didn't think it mattered, my bad (Lansing, Michigan).

Things I've read recommend a sterilized blade for cutting the roots - either boiling in water, or using fire/butane torch. (I'll probably boil; I just have regular BIC lighters, no butane anything) Unless this isn't really necessary? Well boiling is simple and can't hurt, even if it isn't necessary.

Do I do anything with the cut ends of the roots - like apply anything to them, or try to shove them down in the potting medium (a bark of some kind, it looks like)? Or are they fine just floating above the pot once they're cut?

Thanks again!

Do not worry too much about all this sterilising, where the orchids grow in nature is not some sort of pristine (germ free) wilderness. Find some more notes and read....especially notes written for Northern USA.
Much to some dismay, the plants will turn towards the light and lean over and they will grow roots out into the air and into the mixture. New roots whichever way they grow are a good sign. If you describe where you have the orchid relative to windows and such the locals may add to this. A nice centrally heated house in winter is a plus.
A temperature/ humidity gadget is a good investment.

None of the windows in the house have a ledge/shelf, so I have it sitting on the edge of a kitchen counter near a southern-facing window with a sheer curtain. Here it is about noon today

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I haven't been through a winter in this house yet, just moved here a month ago. I should be able to move it further away from the window in winter if it's chilly there, maybe to the dining room table where there's an eastern-facing window (though light will be limited, as there's 2 huge maple trees providing shade not too far outside that window; but it's still bright there).

I believe this will end up being insufficient light this far from a window. Phal's like shade, but they also like bright light and very early morning/late evening sun (as long as they don't burn).
Do not try to force the roots in any way...they'll break.

I can't tell for sure, but it does look to me that you might want to prepare to repot either later in fall or next spring. It depends on bark breakdown..I think it's been in that pot for a while (so I'm guessing).

I don't know if this is of any help, but I sterilize all my clippers in a can or jar of alcohol, between each cut... I keep a large open-mouth jar full of alcohol, in a pail, with my clippers, and dip the clippers between each cut...and put the cover back on the jar of alcohol when I'm done.... I have a large rose-bed and learned to do this the hard way--- several years ago I would just do my pruning with the same clippers, no sterilizing-- until I was hit with canker and it spread like wild-fire, it is spread by cutting, wounds to the cane, cross-canes that scrape each other--etc.also the weather contributes, cool and rainy and overcast... and I lost a lot of my roses--- it was just the straw so to speak that woke me up to my carelessness, an expensive lesson! So now I keep a pail for the roses, and another pail for the orchids, both with the jar of alcohol and their own clippers... and have not had any problems since.... works for me, LOL, sally