The transverse momentum ($p_{\rm T}$) spectra and elliptic flow coefficient ($v_2$) of deuterons and anti-deuterons at mid-rapidity ($|y|<~0.5$) are measured with the ALICE detector at the LHC in Pb-Pb collisions at $\sqrt{s_{\rm NN}} = 2.76$ TeV. The measurement of the $p_{\rm T}$ spectra of (anti-)deuterons is done up to 8 GeV/$c$ in 0-10% centrality class and up to 6 GeV/$c$ in 10-20% and 20-40% centrality classes. The $v_2$ is measured in the $0.8 <~p_{\rm T} <~5$ GeV/$c$ interval and in six different centrality intervals (0-5%, 5-10%, 10-20%, 20-30%, 30-40% and 40-50%) using the scalar product technique. Measured $\pi^{\pm}$, K$^{\pm}$ and p+$\overline{\mathrm{p}}$ transverse-momentum spectra and $v_2$ are used to predict the deuteron $p_{\rm T}$ spectra and $v_2$ within the Blast-Wave model. The predictions are able to reproduce the $v_2$ coefficient in the measured $p_{\rm T}$ range and the transverse-momentum spectra for $p_{\rm T}>1.8$ GeV/$c$ within the experimental uncertainties. The measurement of the coalescence parameter $B_2$ is performed, showing a $p_{\rm T}$ dependence in contrast with the simplest coalescence model, which fails to reproduce also the measured $v_2$ coefficient. In addition, the coalescence parameter $B_2$ and the elliptic flow coefficient in the 20-40% centrality interval are compared with the AMPT model which is able, in its version without string melting, to reproduce the measured $v_2$($p_{\rm T}$) and the $B_2$($p_{\rm T}$) trend.