Can I use the coalesce operator on integers to chain CompareTo?

No basically, but it would be nice if it did (IIRC, Jon mentioned a similar idea in C# in Depth). You could probably chain conditionals, but I tend to just use:

int delta = Bar.CompareTo(rhs.Bar);
if(delta == 0) delta = Baz.CompareTo(rhs.Baz);
if(delta == 0) delta = Fuz.CompareTo(rhs.Fuz);
return delta;

Not really, ?? only works for null values (reference types or nullable structs)

int i;

i = Bar.CompareTo(rhs.Bar);
if (i != 0) return i;

i = Baz.CompareTo(rhs.Baz);
if (i != 0) return i;

i = Fuz.CompareTo(rhs.Fuz);
if (i != 0) return i;

return 0;

Not supported by the language. But you can write a small helper like this:

public override int CompareTo (Foo rhs)
{
    return FirstNonZeroValue(
        () => Bar.CompareTo(rhs.Bar), 
        () => Baz.CompareTo(rhs.Baz),
        () => Fuz.CompareTo(rhs.Fuz));
}

private int FirstNonZeroValue(params Func<int>[] comparisons)
{
    return comparisons.Select(x => x()).FirstOrDefault(x => x != 0);
}