Is it possible for a distribution to have infinite variance but finite covariance or vice versa? Is it possible to have distributions s.t. one/both have infinite variance, but finite covariance? What about finite variance but infinite covariance?
If so, what are example distributions/what is the constraint on the pdf that makes this true?

The $t_2$-distribution has infinite variance but finite first moment. If you use a constant (which technically is a one-point distribution) as second distribution/random variable, the covariance will be zero, so finite. In general, if one variance is finite and the other one infinite, both finite and infinite covariances can happen.
For two independent $t_2$-distributions, both obviously with infinite variance,
$$
Cov(X,Y)=E(X-EX)(Y-EY)=E(X-EX)E(Y-EY)=0,
$$
not involving any degenerate integrals.
Finite variances and infinite covariance is impossible due to the Cauchy-Schwarz inequality,
$$
Var(X)Var(Y)\ge Cov(X,Y)^2.
$$
