Stats help for categorical data

The data are hospital admissions data for cause X and admissions data for all admissions. If an individual has been admitted for cause X, they are coded 1 (versus 0). If they were admitted for any cause, they are coded 1 (versus 0). If they are admitted for cause X then logically they are also coded for admissions for any cause (so both would be labelled 1). We also have deprivation data (1, 2, 3, 4, 5), gender (1,2) and age category (1-15).
The questions we want to answer are:
Are males/females more likely to be admitted for cause X over any cause? Is the difference for being admitted for cause X over any cause higher in males than in females? Are males more likely than females to be admitted for cause X?
Are those living in any deprivation quintile more likely to be admitted for cause X than any cause? If so, is the difference between quintiles for being admitted for cause X over any cause statistically significant? Are there any quintiles that are more likely to be admitted for cause X?
Are there particular age groups that are more likely to be admitted for cause X over any cause? If there are age groups that are more likely to be admitted, is the difference statistically significant? Are any age groups more likely to be admitted for cause X than others?
So far, we have decided to do chi-square to look at expected and observed values for gender, deprivation and age categories for 1) all admissions 2) admissions for cause x.
But for example, it's highly likely that males are more likely to be admitted for cause X than any admissions, whereas this is not the case for females. How do I test if admissions for cause X in males vs. admissions for any cause is significantly different from the same scenario in females?

I hope this makes sense and thanks in advance!