We investigate the zero-temperature glassy transitions in the square-lattice +/- J Ising model, with bond distribution P(J(xy)) = p delta(J(xy)-J) + (1-p)delta(J(xy)+J); p=1 and p=1/2 correspond to the pure Ising model and to the Ising spin glass with symmetric bimodal distribution, respectively. We present finite-temperature Monte Carlo simulations at p=4/5, which is close to the low-temperature paramagnetic-ferromagnetic transition line located at p approximate to 0.89, and at p=1/2. Their comparison provides a strong evidence that the glassy critical behavior that occurs for 1-p(0)<p<p(0), p(0)approximate to 0.897, is universal, i.e., independent of p. Moreover, we show that glassy and magnetic modes are not coupled at the multicritical zero-temperature point where the paramagnetic-ferromagnetic transition line and the T=0 glassy transition line meet. On the theoretical side we discuss the validity of finite-size scaling in glassy systems with a zero-temperature transition and a discrete Hamiltonian spectrum. Because of a freezing phenomenon which occurs in a finite volume at sufficiently low temperatures, the standard finite-size scaling limit in terms of TL1/v does not exist; the renormalization-group invariant quantity xi/L should be used instead as basic variable.