A cube of ice dropped into water, seen from the side: how much sinks, how it bobs, what the water pushes with, and what happens to the water level when the ice melts.
The science
Archimedes. A floating body displaces its own weight of fluid: rho_f g V_sub = m g, so the submerged fraction is rho_object / rho_fluid. Ice (917 kg/m3) in fresh water (1000 kg/m3) sits 91.7 % under; in sea water (about 1025 kg/m3) it sits 89.5 % under.
Where the force comes from. Pressure grows with depth, p = rho_f g z; integrated over the bottom and top faces the difference is the buoyant force rho_f g V_sub.
Bobbing. Displaced a little, the cube feels a spring-like force with stiffness k = rho_f g A (A = base area), so it oscillates with period T = 2 pi sqrt(m / (rho_f g A)), damped by the water.
Melting. The ice displaces m / rho_f of fluid but melts into m / rho_water of fresh water. In fresh water these are equal and the level does not change; in denser salt water the melt volume is larger than the displaced volume, so the level rises slightly.
Simplifications. Calm water, rigid upright cube, no added mass or wave drag beyond a fixed damping ratio, no thermal effects or salt-ice mixing, a load can be added on top.