1. Where would the center of mass be of the Pluto-Charon system?
The exact answer depends on where you add the mass. It would be at the proportional average of the distribution of the mass, as will all other connected bodies. But the change would be very small.
The mass of Charon is 6.5 orders of magnitude smaller than Pluto , so it’s less than one millionth the mass of Pluto. If you add a new Charon’s mass to Pluto, you change things by less than one one-millionth. If you just move Charon to Pluto, that’s not changing the mass of the system – just moving its center slightly.
2. What would happen to the moons?
Almost nothing.
The moons of Pluto would have their orbits altered very very slightly.
If you’re just moving Chaon onto Pluto, Charon was already inside the orbits of the other moons , so you’re only very slightly changing where the center of mass is.
If you magically added mass to Pluto without removing Charon, the effect would be to slightly increase the gravitation but only extremely slightly, so the relative orbits would stay roughly the same but a tiny bit faster and closer.
3. Would Pluto remain a dwarf planet?
Up to whoever invents the categories and their definitions, but yes.
4. Would any other orbits be affected?
Not materially.
If you want to see for yourself, you could try Universe Sandbox