The answer to question 1 will depend on your research question, and who the audience are for the result.
If your research question points to talking about differences in b based on the profile of A, then that will obviously help frame your summary. In an epidemiological study, even if you're not sampling based on A (independent variable as exposed/unexposed status) it would still make sense to use this classification as an independent variable [exposure] and the continuous variable as a dependent variable [outcome]. It sounds like you already know the answer to this.
You should also consider how you might interpret the result in terms of presenting the results to others (and interpreting it yourself). A continuous-variable-as-dependent variable [outcome] model would have a mean difference (or similar) as one summary; a dichotomous-variable-as-outcome model would have an odds ratio (ratio of increased odds per one unit of the continuous variable, which could be scaled to give e.g. relative increase per five kilos of additional weight for likelihood of type II diabetes.)
My experience from consulting settings and explaining this to people is that the former (difference in means) is generally more easy to explain to other people than the latter (odds ratio per one unit difference of continuous independent variable.)
For your question 2, if you want to run a multivariable model, where you're controlling for covariates, then it will help to choose dependent/independent variables at the start. It's probably best to stick with the same method from univariate to multivariable analysis, rather than changing between the two approaches, just from ease of explanation.
Final note on this latter point: from a hypothesis testing perspective, a logistic regression with a continuous independent variable [exposure] and [single] dichotomous dependent variable should return the same p-value as an unpaired t-test assuming unequal variance with the variables reversed (from memory -- I'm not entirely sure if this is always true though.)