One must imagine Pluto happy.

The largest ring of Jupiter is just about 129 000 km of radius. The nearest Galilean moon to jupiter has a semi-major axis of 421 800 km, so the rings aren't in any of their orbital neighborhoods.

Beside, the largest of them, Ganymede, is more massive than Mercury.

But you're right that not all the moons would be either planets or dwarf planets, many would be asteroids.

From Wikipedia

Of Jupiter's moons, eight are regular satellites with prograd and nearly circular orbits that are not greatly inclined with respect to Jupiter's equatorial plane. The Galilean satellites are nearly spherical in shape due to their planetary mass, and are just massive enough that they would be considered major planets if they were in direct orbit around the Sun.

The moons would almost certainly fall into Jupiter or be thrown from its orbit if itsuddenly gained enough mass to become a star.

The sun and Jupiter are pretty close in terms of density, and Jupiter would need to get at least an order of magnitude heavier to start fusion. I think it's just a coincidence that the outward pressure of the sun's fusion makes these numbers roughly line up.

Thirteen Jupiters seems to be a commonly-given lower limit for fusion, so let's go with that. To increase mass by thirteen times while maintaining density (and assuming the whole thing is a perfect sphere, which it obviously isn't), Jupiter needs to increase its radius by a factor of about 2.35. This increases its equatorial radius to about 168,000 km, which does swallow up the three innermost moons, but leaves the four big ones alone

They’re in an orbit that works with jupiters current mass. If Jupiter gained enough mass to trigger fusion then I would think that many of them would with fall in or be slung out of orbit.

I’d be interested to see if anybody has done the math though to see how that would all play out.

Aside from the orbit change mentioned, the huge increase in stellar radiation would certainly blow much of the lighter elements including water away. The core and some residuals that might remain on the far sides would be all that's left.