Covariates in regression models Should covariates be included in regression analyses if they are correlated with the dependent variable or if they are correlated with the predictor variable/s. Alternatively, should they be included as a result of past (fairly robust) findings that they are significantly related to the outcome and/or predictor variables?
 A: Correlation with the dependent variable is a definite plus (especially for linear regression where there are close links between the coefficients and covariance with the dependent variable).
Correlation with the other covariates/predictors is somewhat more subtle and depends on your goal. Generally, it is considered good practice to include as many variables as feasible at first (especially ones that you have some reason to believe could be relevant from previous research or the likes), and then try to optimize with some criterion (e.g. AIC or simple likelihood ratio tests) or through some optimizing algorithm (LASSO etc.).
I make an exception for perfectly correlated variables: there is no use in leaving them both in.
You should always be careful to leave covariates out, though, as leaving the wrong one(s) out could bias your coefficient estimates!
Maybe you can ask your question somewhat more explicitly? Specify what are your goals in your research, then we may be more able to give specific advice.
A: This really depends on the scientific question being asked.  If you are interested in if there is a relationship between x1 and y, then do the regression between x1 and y.  If you are interested in if x1 helps predict y above and beyond the effects of x2, x3, etc. then you need to include the other x's in the model.
For example:  suppose that Y is length of stay in the hospital and x1 is the dose (amount) of medication for a given set of patients.  Now you may want to adjust for the severity of illness measured at the time of entry into the hospital/study and the treatment starts just after the measure of the severity, then you would probably want to include severity in the model.  But if your severity of illness measure is the most severe the illness gets during their stay and the medicine reduces the severity and therefore shortens the length of stay, then including severity of illness will hide the effectiveness of the treatment rather than give any useful information.
Before doing a regression it can be helpful to draw out a path diagram, write out the names of all the variables that could go into the regression, then draw arrows where you know, or strongly believe that there are relationships and how the causality most likely goes.  Then draw an arrow or arrows in another color or style to show what relationships you think might exist but don't know and want to test.  Considering this diagram can be useful in thinking about what models make the most sense to fit and compare to each other.
