Yes, logistic regression is suitable, assuming you are treating the binary variable as the DV and the continuous variables as the IVs. Not sure about other methods.
The answer depends on what you are trying to do. If you want to model odds ratios, then logistic regression is fine. If you want to model incidence ratios, then a log binomial generalised linear model is better. If you are unsure, best seek help!
Identify whats your dependent variable and independent variables first. In the logistic regression model, you may include variables which are continuous as well.
For logistic regression, only your dependent variable should be categorical, if it has 2 category then Binary logistic regression, for more than 2 categories it is multinomial logistic regression.
There is no any restriction on Independent variable, it can be Categorical as well as continuous.
Yes, the logistic regression model can accommodate both categorical and continuous independent variables while your dependent variable must be categorical with two (binary) or more (multinomial) levels
I think it is not common in public health research, but it is possible to do that. However, the interpretation is a bit different and makes less sense sometimes. For example, there is an increase x% of [outcome variable] , every unit of [independent continuous variable ] increase. If possible, you can convert the noncontinuous to ordinal categorical variables.