What are the minimum depth circuits possible for addition and multiplication of two n-bit numbers using just AND and XOR gates? I read somewhere that we can achieve constant depth for addition if we have an OR gate. Can I achieve that using XOR gates?
You can simulate an OR gate using a constant number of AND and XOR gates:
$$x \lor y = ((x \oplus 1) \land (y \oplus 1)) \oplus 1.$$
Consequently, anything you can do in depth $d$ using AND and OR gates, can be done in depth $\le 3d$ using AND and XOR gates.