SUMMARY: It's Tika's tooth.
At least for this time around, I've figured out why Tika suddenly acts miserable and sore when playing tug-of-war and becomes subdued and won't play. Why bending her neck to the left hurts. Why she'll be curled over to lick/nibble certain parts of her body and suddenly yelp and look miserable for a couple of minutes.
It's not her neck. Not her back. I just happened to look down at her from the correct angle just now, and noticed her left cheek (opposite my hand in this photo). I tried to put my finger inside to pull the cheek out to look, and she shrieked. Just touching the outside makes her yelp. I can't tell so much from wrestling her mouth open and looking inside, but that section of her gums does look a bit inflamed. Still, it looks like it's actually the skin on her cheek that's swollen--like the lump is NOT in her gums; when I do manage to pull the side of her mouth out, the lump seems to come with it. I've never had a dog with tooth problems or abscesses before, though, so don't know what to expect.
I don't think that this explains why, 4 weeks ago, she couldn't jump into her crate without yelping. But it fits everything else. I've got an appointment with the vet for tomorrow morning. I feel an ominously large vet bill coming.