Is infinity the reciprocal of zero/is zero the reciprocal of infinity?

It depends on the number system you're using.

If you're using the real numbers or the complex numbers, then zero has no reciprocal. In other words, $1/0$ is an undefined expression. Also, in these systems, there's no such number as infinity. In other words, $\infty$ is an undefined symbol.

If you're using the projectively extended real line or the Riemann sphere, then the reciprocal of zero is infinity, and the reciprocal of infinity is zero. In other words, $1/0 = \infty$ and $1 / \infty = 0$. (Note that the reciprocal of infinity is exactly zero, not infinitesimal. None of these four number systems contain any infinitesimal numbers.)

Out of these four number systems, the first two (the real numbers and the complex numbers) are much more commonly used than the last two (the projectively extended real line and the Riemann sphere). So much so, in fact, that we usually say "division by $0$ is undefined" and "infinity is not a number" without clarifying which system we're using.

The reason that the first two systems are more commonly used is that these two systems are fields, and the other two are not.