  Aims. We use different tracers, such as Ha, ultraviolet (UV), and infrared
(IR) emissions at various wavelengths, to study the dust and star-formation
(SF) conditions throughout the disk of M33. Methods. We derive the radial
distribution of dust, of the old and young stellar population using Spitzer and
GALEX data, complemented by ground-based optical data and available surveys of
atomic and molecular gas. We separate the contribution of discrete sources to
the IR brightness from the diffuse emission. Results. At 8 and 24 um, discrete
sources account for 40% of the IR emission in the innermost 3 kpc, and for 20%
further out. We find that stochastic emission from very small grains in the
diffuse interstellar medium accounts for only 10% of the diffuse 24 um
emission, and that dusty circumstellar shells of unresolved, evolved AGB stars
(carbon stars) are a viable alternative. The 8 um profile suggests that PAH
emission declines faster with radius than the dust continuum. In annular
regions, 0.24 kpc wide, we find a mean extinction value for stellar continuum
AV = 0.25 mag with a weak dependence on radius, consistent with the shallow
metallicity gradient observed. Dust opacity derived from the 160 um emission
decreases instead by a factor 10 from the center to edge of the SF disk.
Conclusions. Using extinction corrected UV and Ha maps we find the global SF
rate in M33, over the last 100 Myr, to be 0.45 +- 0.10 M yr-1. FIR (far-IR) and
TIR (total-IR) luminosities can trace SF even though a high conversion factor
is required to recover the effective rate. If carbon stars are powering the
diffuse 24 um emission in M33 this can trace star formation 1 Gyr ago and
provide a more complete view of the SF history of the galaxy. [abridged]
