How can we have "floating" and convection currents in a non-liquid mantle?
Is the mantle solid?
It's all a matter of timescales. The mantle is undoubtedly solid (except locally in the uppermost regions where melting can occur) but on a long enough timescale it can display plastic behaviour under high temperatures and pressures, with a viscosity on the order of $\text{~}\mathrm{10^{21}~Pa~s}$ (based on estimates from isostatic rebound after melting of ice sheets). Think of plasticine - it is obviously solid but it deforms easily under stress. This is how the mantle behaves on long timescales at high temperatures and pressures. On short timescales the mantle is a brittle solid and this is what allows it to transmit seismic S-waves.

Why does the crust 'float' on top of the mantle?
The answer to this is quite simple - the crust is less dense than the mantle and so on the long timescales, where the mantle is ductile, the crust does not sink into the mantle (except at subduction zones but that's another story). Granitic continental crust has an average density of around $\mathrm{2.6~g~cm^{-3}}$, basaltic oceanic crust is around $\mathrm{3~g~cm^{-3}}$, and the upper mantle is around $\mathrm{3.5~g~cm^{-3}}$.

How does convection work in the mantle? 
The details of mantle convection are a subject of active research but all geologists agree that the mantle does convect vigorously and that this is the major form of heat transport from the core to the crust. If you calculate the Rayleigh number for the mantle you come out with $R_a~\text{~}~10^6$ which indicates that convection should be vigorous. On the other hand, the thermal diffusivity of the mantle is very low ($\text{~} \mathrm{10^{-6}~m^2~s^{-1}}$) and so conduction is not the dominant form of heat transport.