How do you keep small mosaic pieces from popping off during the bisque fire?

Leather hard is actually a reasonable time to do this, so I wouldn't move all the way back to soft slab. The problem is more likely that the tiles and the body are drying at different rates. Little pressed-in bits dry way faster than the big mass of the mug, so they shrink first and pull away at the edges. What fixed this for me was wrapping the whole thing loosely in plastic for a day or two after attaching, so everything equalizes before it goes to bone dry. Slow and even drying matters more than when you attached them. That said, if your clay bodies have even slightly different shrink rates, bone dry and bisque is where the mismatch shows up. Attaching a bit wetter, closer to soft leather hard, gives them more chance to shrink together instead of fighting each other. And make your slip from the same body as the mug, not the tile clay. I also score the tile backs properly, not just a quick scratch. A drop of vinegar in the slip helps it grab if you're attaching on the drier side.

One other thing that lifted my success rate a lot: pressing firm can actually work against you. If you really bear down on a tile at leather hard, you distort the edge and create a little ridge of squeezed-out slip that cracks off later. Firm but even pressure, then compress the edge with a fingertip or a small rib instead of mashing the tile down.

For the ragged edges, I clean them up at leather hard while it's still trimmable. A loop tool or even just the edge of a metal rib takes the burrs off in seconds and it's way easier than sanding after the bisque. If you leave the bumps, glaze will pool in the seams and soften them somewhat, but any sharp burrs will still sharpen up in the fire and catch on lips and hands. Flush at leather hard, then a light sponge at bone dry, and you'll barely sand at all.

Gotta push back on the slip advice though. Telling them to make slip from the mug body seems backwards to me, since the tiles are the ones lifting, not the mug. When I did inlay stuff I mixed slip from the tile clay and my failure rate dropped hard. My theory is the joint ends up matching the piece that wants to move most, so it moves with it instead of tearing. Could be total coincidence and my drying was just slower that month, but a couple hundred tiles later I'm not switching. Would be curious if anyone's actually tested both ways on the same mug.

I gave up trying to get mosaic tiles perfectly flush before firing. Now I set them a hair proud on purpose and flatten them after bisque with a diamond sanding pad, wet, on the wheel head so I can spin it. Takes a couple minutes per mug and the edges come out dead flat, way faster than chasing every burr with a loop tool while the clay's still soft. The dust is the annoying part so I keep it dripping wet and work outside. Do you find your tiles sit proud or sunk when they come out of the bisque? Mine always end up a touch low in the middle, which is why I stopped fussing pre-fire.

I've only tried stuff like this on flat coasters, not mugs, so grain of salt, but the thing that bit me was the tinted clay bleeding color into the darker body around each piece, mostly with the reds and oranges. Do yours stay put after glaze or do you get little halos? Also wondering if your rim failures happen more where the wall curves. On my coasters everything held fine, but the one time I tried it on a curved form the corners of my pieces popped off, so I always assumed curvature was the enemy and never tried again.

What changed everything for me on inlay work was carving a shallow seat for each tile instead of just pressing them onto the surface. I use the small end of an old loop tool to cut a tiny recess, then set the tile in with slip so it's sitting in a little socket. Even when the joint wants to separate, the clay wall around it physically holds the tile in, so nothing pops off at the rim. Takes a few extra seconds per tile but my loss rate basically went to zero. Curious if that would work on your curved walls or if the seats only behave on flat sections like the coaster person was saying.