I understand the Steenrod algebra for $\mathbb{F}_{p^n}$ both from classical calculations and a past question, but I'd like to ask about $H\mathbb{F}_{p*} H \mathbb{F}_q$ for $p$ and $q$ different primes. Does anyone know the answer to this or a reference I could check out?
It would also be nice if I could know what $H\mathbb{F}_{p^n*} H \mathbb{F}_{q^m}$ is in general but this seems significantly harder.
If $p$ and $q$ are distinct primes, then ${H\mathbb{F}_p}_* H\mathbb{F}_q = 0$. This follows from early work of Serre on the cohomology of Eilenberg-Mac Lane spaces. The $p=2$ case, for example, is stated (with references to a few other papers of his) in the second paragraph on p. 207 of "Cohomologie modulo 2 des complexes d'Eilenberg-MacLane" (https://link.springer.com/article/10.1007/BF02564562). (Well, he's dealing with cohomology rather than homology...) You can prove this for spaces using Serre's theory of classes of abelian groups. For spectra, you can note that the multiplication by $p$ map is an isomorphism on $H\mathbb{F}_q$ but is zero on $H\mathbb{F}_p$, and therefore is both zero and an isomorphism on $H\mathbb{F}_p \wedge H\mathbb{F}_q$.
The same arguments should work with $p^n$ in place of $p$.