А model without balanced growth

For the first issue, there is no general answer as we do not know the data-generating process. Thus, it is your decision whether you think that i) there will be balanced growth in the long run and the current behavior is just convergence to this point or ii) that non-balanced growth is a feature of the data that you need to model. Section 4 of Fernald (2014): “Productivity and Potential Output Before, During, and After the Great Recession” at frbsf.org/economic-research/files/wp2014-15.pdf might be informative to guide your thought process.