I will expand what I wrote on my comment and I'll correct my terminology, following AdamO.
Regarding the first question, I would answer no for two reasons: a) if $A$ or $B$ are continuous, you have to look in terms of increase in each variable if you want to talk about ratio of odds ratios; b) if $A$ and $B$ are categorical, it would only make sense if you have multiple categories (at least 4 if $A$ and $B$ are ate least dichotomous) and the ratio of odds ratio have to be interpreted in terms of the reference categories (I will talk about the categorical variables later).
With respect to the second question, if both $A$ and $B$ are continuous (as your comment indicates), I agree with your interpretation:
an increase in the interaction term ($A \times B$) by one unit of
measure increases the odds ratio of "success" by a factor of $1.5$.
The third question says more or less the same thing as the second one but with a reorganization of the terms. Since $A$ and $B$ are continuous I agree with you on:
if $A$ increases by one unit, then the odds ratio increase by a factor of $\exp(𝛽_1+𝛽_3 B)$.
Regarding on the way I interpret the interaction terms, I usually begin by looking at the isolated terms to understand the effects of each variable, then I look at the interaction terms.
If both variables are continuous, I would look at sign of $𝛽_1+𝛽_3 B$ for low $B$ and for high $B$. If the signal changes it means that for low $B$ the effect of $A$ is in one direction, while for high $B$ the effect is on the opposite direction. If the sign does not change, the value of $B$ that makes $𝛽_1+𝛽_3 B$ closer to zero would mean that for that value of $B$ the effect of $A$ would be less visible, while for the value of $B$ on the "other side" of its mean, the effect of $A$ is "stronger".
If one or both variables are categorical, the interaction term has to be interpreted having in consideration the reference category of the variables, which usually makes our life hard because we will have to take in consideration the degrees of freedom of the interaction term. In general terms, if both are categorical, we would say something like the effect in the category $a_j$ of $A$ is smaller or larger among the individuals that belong to category $b_k$ of $B$.
See also other questions on interaction terms in conditional logistic regression and in logistic regression and a related question.