I'm pretty sure that the police presence at the Synagogue was initiated in response to the 2015 attack there (https://en.wikipedia.org/wiki/2015_Copenhagen_shootings), but is probably ongoing because of continued security concerns after more recent attacks elsewhere.

The police incident we witnessed appears to have started in Tivoli. This is the best article I could find (at least in English): http://www.express.co.uk/news/world/812426/Copenhagen-shooting-gunman-updates-Tivoli-Gardens It overstates the "lockdown", which really was only a few blocks around the incident. Live pretty much went on as normal in most of the city.