Why can molecules accept only quantitated energy(photons), but can have any temperature(can accept any amount of heat ot have any temperature?
Complicated. The quick and dirty answer is that the molecules still have quantized energy states in the thermal energy region (IR), but the states have a high density, particularly in condensed phases. Broadening of individual peaks makes the absorption seem "continuous". Translational energy of molecules is also basically unquantized... or, the states are so close together that they are practically continuous. A more detailed answer would include discussion of phonons
Why have molecules continous spectra?(assuming one photon with specific wavelength and energy produce single specific electronic transition, so it woud be logical that absorb only specific wavelengt or few of them)
Spectral transitions never are monochromatic. There is always spectral breadth. At the most fundamental level, broadening of peaks is due to the uncertainty principle, whereby electrons spend a finite amount of time in a state, therefore there is a limit to the degree to which the energy of the state can be known. Lifetime broadening results from this uncertainty. But, the true spectral breadth is usually larger than the uncertainty-limited value. These types of broadening can be do to lots of things, such as a time-dependent variation in molecular structure that causes slight differences in the energies of absorbing states, the translational velocity of molecules (doppler effect, basically), etc.
You can distinguish between the different causes of broadening by the lineshape profile.