Without having more information about what you are trying to accomplish or what type of data you are using, I suspect you will run into an issue here. Testing an independent variable requires that all other data is fixed (non-changing), for the test variable to be truly independent. In your case, this would not be possible since you would have more than one independent variable. At first glance, I would suggest duplicating your data model and applying only one independent variable in each, while keeping the cross-sections the same. If you have all 6 or 7 independent variables in the same data model, it would likely be automatically treated as dependent variables instead of independent.

Well, adding more independent variables has its penalty which is mainly the R-square adjusted.

Also, being as parsimonious as possible is quite necessary for model building. You could run a pre-analysis to confirm the significance of the independent variables you want to add. However, it is advisable to make the whole variables not more than 6.

Thank you for the question asked. For a consistent and reliable estimation, the assumption is that the number of observations should be more than the variables in the model. Therefore, if this assumption is realised, then it is possible to analyse the model.