Run regressions separately for each dependent variable.

If you have a series of univariate questions, estimate a separate univariate model for each outcome variable.  If you have a multivariate question, estimate a single multivariate regression model.  See Huberty & Morris (1989) for info & discussion. 

Note that univariate and multivariate describe the number of outcome (dependent) variables in a model, not the number of explanatory variables.

HTH.

https://pdfs.semanticscholar.org/ee6c/77c99c8e4530d0cccaedf85ed525fb22a02d.pdf

Yes, you can use more uni- or multivariate regressions. However, if you have one dependent variable with more than two categories, you can use multinomial or ordinal regression.

https://www.analyticsvidhya.com/blog/2016/02/multinomial-ordinal-logistic-regression/

The best possible answer is to run the regression seperately for each Depedent variable as suggested above.

The selection of regression depends on the characterstics of dependent variable.

If the dependent variable is categorical and having two category, go for binary logistic regression. and if the dependent variable having more than two category then go for multinomial logistic regression.

But if the dependent variable is continuous then go for linear regression or any other continuous regression as appropriate with your data and objective.

yes, it is possible. What you're interested is is called "Multivariate Multiple Regression" or just "Multivariate Regression". I don't know what software you are using, but you can do this in R.

Here's a link that provides examples.

http://www.public.iastate.edu/~maitra/stat501/lectures/MultivariateRegression.pdf

the so called Partial Least Squares (PLS) might be useful - googlize it to get descriptions and uses of the method

PLS and SEM is suitable in this case