Converting a digital circuit to two layers of OR and AND gates
Yes, it's true.  You can use  conjunctive normal form to express the circuit in two layers, where the first layer has OR gates and the second layer consists of $m$ (multi-input) AND gates.  However, there's a catch: you have to have a way to complement (negate) some of the inputs to the OR gates for free.  If you want to count the OR gates as a separate layer, then it takes three layers; if you imagine that your OR gates actually allow you to complement any subset of their inputs (e.g., the gate can compute $x \lor y$, $x \lor \neg y$, $\neg x \lor y$, or $\neg x \lor \neg y$), then it's just two layers.

Or, you can use disjunctive normal form to build a two-layer circuit where the first layer uses AND gates and the second layer contains $m$ (multi-input) OR gates.  Again, the same catch applies: you need to be able to complement the inputs to the AND gates "for free".

You can use disjunctive normal form to build a circuit where at most one input to any OR gate will ever be true at a time.  This means that you can replace the OR gate with an XOR gate without changing its functionality.  Therefore, yes, you can built a two-layer digital circuit where the first layer has only AND gates and the second layer has only XOR gates -- assuming you can complement some of the inputs to the AND gates for free.

For "XOR-of-ANDs", you can actually write any N-to-1-bit circuit as an XOR of ANDs, without any complementation at the input, if you allow the final XOR to optionally include an additional input that's held at the constant 1 (or, equivalently, if you allow the final gate to be either a XOR or a NXOR).  In particular, given any boolean function $f:\{0,1\}^N \to \{0,1\}$, you can write it as a multivariate polynomial $f(x_1,\dots,x_N)$ modulo 2.  Now $x^2=x \pmod 2$, so since everything is modulo 2, that means that you can eliminate all powers of two or higher: every multivariate polynomial can be expressed in the form

$$f(x_1,\dots,x_N) = \sum_S c_S \prod_{i \in S} x_i \pmod 2,$$

for some constants $c_S$, where $S$ ranges over all subsets $S \subseteq \{1,\dots,N\}$.  (For example, we might have $f(x_1,x_2,x_3) = x_1 x_2 + x_2 x_3 + x_3 + 1 \pmod 2$ as an example of such a polynomial.)  Now note that product modulo 2 is AND, and sum modulo 2 is XOR, so we get a XOR-of-AND representation of any boolean circuit.

As your colleague mentioned, in the worst case these circuits might need approximately $2^N$ gates, so this can be highly inefficient in some cases.

I don't know if it is possible to write every circuit as an AND-of-XORs (with some complements or constants somewhere).