A control variable (confounder, potential omitted variable) is a variable you include in the model because you suspect it is confounding the main relationship you are interested in (so it is suspected to be related to both the main independent variable (explanatory variable, predictor, treatment) of interest and to the dependent (outcome) variable.
A moderator is a variable which changes the effect of the main independent variable on the outcome variable, so it interacts with the main independent variable.
A mediating variable is a variable that translates fully or partially the effect of the main independent variable on the outcome.
A covariate is in my opinion unspecified and could refer to either of these.
A variable that is suspected to be related with the outcome variable but not with the main independent variable of interest is a covariate but not a control variable (as it does not control for anything).
A variable that is suspected to be related to the main independent variable of interest but not to the outcome would be an instrumental variable.
To come back to your concrete questions:
since you don't use them it matters little what you called them - alternative predictors (causal factors), unmodelled effects, etc are all options
definitely no controls or covariates, these are just sample selection criteria
zvariable from @ACD's previous comment would be a covariate and not a moderator, since there is no interaction. $\endgroup$