Guys is it safe to keep UPS connected to power supply all time. Will there be any damage to the battery or battery may blowup due to overcharge?
No. Once the battery is charged completely it will shift to intermittent trickle charging or something. It won't blow up.
I always connect my APC (BR1000UX) ups to power. It is working this way for last 4 years, no issue. Switching power on and off for UPS is hard, also i need my DSL modem (WIFI) always ON.