You want to predict a binary (0/1) variable, definitely start out with logistic regression. ANOVA is really a variant on usual linear regression (for continuous response), and uses some assumptions that are definite not true for binary response, like constant error variance.
As for the numbered points, this is a very broad question ... but note that logistic regression is a specific case of glm (generalized linear models), and usually methodology for glm's closely parallels that for ordinary linear regression. This is already widely discussed on this site, so I will just link to some relevant posts:
Assumptions.
Interpreting residual diagnostic plots for glm models?