Is it permissible to engage in congregational tasbeeh after the prayer, and does the failure to straighten the rows prevent the prayer from being valid?

You are not obligated to go to the aforementioned mosque, but if you wish to go, the lack of straightening the rows and the communal recitation of glorification (tasbeeh) are not impediments, as long as they perform the prayer with its pillars and obligations. You can also leave to complete your recommended remembrances (adhkar) after the prayer.