ANOVA is the equivalent of running multiple t-tests. So, the t-test is a special type of ANOVA.
T-test can be used when we have only two populations to compare their means, while ANOVA is used to compare means between two or more groups. So for two groups, you can use both t-test and ANOVA, and the results would be the same.
Any two variables data, t-test ,chi square and two WAY ANOA could be used. However, if observations are signs only such as + and- or yes or no, sign test is the best.
For comparisons of two groups, the two-sample (i.e., independent-samples) t-test is sufficient; however, the exact answer of your question depends on how you measure anxiety, depression, etc. Are the scores created as interval-scale variables or ordinal variables? If they are interval but not normally distributed, you will have to look for a non-parametric test. For an ordinal scale, you can use a chi-square test or something similar.
If you want to compare between two groups as male and female you can go through parametric tests like t-test or chi-square, independent-sample tests, if your data fulfill the assumptions of the parametric tests. and the other hand as a non-parametric test like the Mann-Whitney test, Rank sum test, etc. you can also for the differences.
Depression Anxiety and stress are all latent variables. Multi-group analysis in CFA to make sure the scales are invariant across the groups. Then independent sample t-tests to check differences.
one way Anova is the best. if you consider sex as covariate you can use Ancova too.if the levels of depression are 2 t test also can be used.at the end the best guidance depends on your goals of your research