Is irreducible error reducible? Since the irreducible error is like a container where we put what we are not able to use in order to build our model, is it possible that in the future this error can be reduced. Maybe we discover unknown theorems, or we improve our algorithms, otherwise we build better measurement tools. So in fact the irreducible error is irreducible only at the moment.
Is this correct ?
 A: The omitted variable bias is not an irreducible error, but a model misspecification. Omitting a variable that belongs in the true model is a reducible error. There are different ways to account for omitted variables (when you don't have access to the 'true variable'). When omitting a variable your model will be biased and will be misspecified. Your coefficient won't be correct. 
This gives an easy explanation
https://www.albert.io/blog/omitted-variable-bias-econometrics-review/
An irreducible error is an error that you get not because your model is not correct, but because of the data you have. 
A: You're talking about a very special case of omitted variable bias issue. Suppose the true process has k variables, but your model accounts only for k-1 variables. This is omitted variable case. However, you seem to consider the case where this omitted variables is not correlated with the first k-1 variables, because only in this case the estimates of k-1 variables do not change when you add the last variable.
So, answering your question, of course the error will decrease if you added a variable to the model that had to be in it in the first place. However, the late addition will not only change the error estimates, but also the coefficient estimates unless the variable is not correlated with others, which is rarely the case.
