Two real time signals convolving
Convolution is what happens when a signal passes through an LTI system.
$$y(t)=x(t)\star h(t)=\int_{-\infty}^{\color{red}\infty}x(\tau)h(t-\tau)d\tau$$
The keyword in this question is real-time. The associated keyword in the answer would be weather the LTI system is causal or not.

If the system is causal, then $h(t)=0, \ \forall t&lt;0$, and therefore $h(t-\tau)=0,\ \forall \tau&gt;t$ which means
$$y(t)=\int_{-\infty}^{\color{red}t}x(\tau)h(t-\tau)d\tau$$
It means we only need the inputs only from $-\infty$ up to the present time $t$ which is all available for a real-time implementation.

So the convolution can be implemented in real-time if the system we implement is causal.

To conclude and generalize this idea for two signals, if at least one of the signals are right-sided (which is practically the case), then the real-time implementation is possible.

Regarding your other question, it really depends on the delay performance of your system.