Interpretation of logistic regression summary I'm having trouble understanding how to interpret the results in this particular situation.
The research topic is to understand how different cat species (Cats 1-3) get infected by a specific parasite.

In the table above, the base category is Cat1. The p-value for the intercept is high, so we cannot conclude Cat1 has a high/low probability of infection. But, we can say that Cat2 has significantly lower odds of infection than Cat1 since the odds ratio is exp(3.52) = 34 (approx.). In this situation what can I say about the odds (or probability) of infection for Cat2? (Although Cat1 has higher odds than Cat2, we can't say if Cat1 has a high/low probability of infection).
This part seems confusing to me, so any help would be appreciated.

You're right that there is uncertainty about whether the log-odds for $\text{Cat1}$ are greater than or less than zero, but we do have an estimate for the log-odds of $\text{Cat1}$: $-0.19$.
Then we have an estimate for the log-odds of $\text{Cat2}$, relative to $\text{Cat1}$: $-3.52$.
Consequently, the estimated log-odds for $\text{Cat2}$ are $-0.19 - 3.52 = -3.71$.
By similar logic, the log-odds for $\text{Cat3}$ are $-0.22$.
To get the odds, we can take $\exp(\text{log-odds})$.
$$
\text{Cat2 odds: }\exp(-3.71) = 0.0245\\
\text{Cat3 odds: }\exp(-0.22) = 0.803
$$
