If somehow a bacterium, let's say E. coli, is engineered in such a way that it loses the formation of ftsZ ring (or other ways that inhibit cell division), and if other conditions are optimal for bacterial growth, then will the bacterium grow without cell division? If so, how long will it live? And how will it age?
First of all, the growth of a bacterium is associated with an increase in cell biomass.So if a bacterium is restricted from replicating it does not grow. Coming to your later parts of the questions about how long will it live and will it age; Yes, a bacterium would age irrespective of its ability to replicate.According to The Free radical hypothesis, all organisms age due to random deleterious events, oxidative damage by ROS produced during normal metabolism. It has also been found that cells undergo irreparable damage which can also lead to cell death eventually.
A complete review on bacterial ageing: http://www.ias.ac.in/article/fulltext/reso/017/04/0347-0364