Is the order of a number $\mathrm{mod}N$ always smaller than $N$? This is probably very obvious, but I can't convince myself of it. Given $x<N$ coprime integers, is it always true that the smallest integer $r$ such that $x^r-1=0\,\mathrm{mod}N$ is such that $r<N$?

Yes. The least (positive) such integer is in fact smaller or equal than $$\varphi(N)=\#\{y\in\{0,\cdots, N-1\}\,:\,\operatorname{gcd}(y,N)=1\}\le N-1$$
by Euler's theorem.
