Is it safe to say that a XOR with an unknown input provides a random output?
I am wondering if you have an input $x_{1}$ and you XOR it with an unknown random second input $x_{2}$, is it safe to say that the resulting $x_{1}\oplus x_{2}$ will be random ?

Yes, this is literally what proofs of the security of one-time pads come down to.  Informally it comes down to the following observations:

If $x_2$ is random, then it means that each of its bits is 0 or 1 with equal probability and independent of all other bits in $x_2$;
If $x_2$ has those properties, then $x_1 \oplus x_2$ also must.

And this really comes down that after the XOR, for each bit in $x_1$ you get with equal probability, either the original bit from $x_1$ or the flipped bit, and that you XOR all of the bits independently of all other ones.