In principle, you shoudl not remove any variable.

Nevertheless, you might want to apply one criterion according to which you should keep or remove variables (e.g., LASSO, looking at R2, etc.).

Then Should I accept this model with insignificant constant?

In principle, I would do so.

Yes you can. However, certain factors could lead to such results. 1) Make sure to select correct series: are they stock variables (for instance, GDP) or flow variables (growth rate). Make sure you have stock variables. Cointegration between a flow variable such as interest rates with a stock variable such as the GDP is economically not feasible. 2) the dependent variable should be integrated of order one, I(1). This can be relaxed for the independents, but no I(2) or over integration degrees are allowed.You did not mention which series you are using, so I mentioned these to make sure but we can just skip to the following:

Yes, try the none option, it is not obligatory to use a constant term though it is commonly preferred. Are there any other factors for such a result? Do you control for outliers or structural changes? Do it and see if the constant term is still insignificant. Be careful not to add too much dummy variables. In the long run equation 1 dummy is more common, a dummy for 2009 for example. If you need to control for more outliers or innovative breaks, add them to the short run section as the deterministic part. Then, check for the constant term.

Compararlas los resultados y suponer que debo seguir para asegurarme de la eliminación de la variable o el modeló

Thanks dear all for answers and nice cooperation.

My advice to you would be not to remove the insignificant constant as it would not harm your other variables.