I know for some essential stuff ( fraud reporting/card blocking) services available 24X7 but for general stuff (which may be important for the user) services are available only till 5pm/6/8pm max.
None that I am aware of. Niyo Gobal provides 24x7 support but for most banking related queries I have been asked to reach them during business hours
One of them is PNB. I think canara bank also will be in the list.
I got two virtual rupay yes bank credit cards (shared limit with my primary card)
As per my analysis, i applied for the first one via yes bank website & second one applied via application on the same day & same card… After few days Both virtual rupay card got approved…
Very interesting case 
Kotak
So far my experience is good. Sometimes, call gets connected to dumb customer care persons. Depends on our luck. @Abhinav_Goyal
Sbm india customer support 24X7 without any IVR drama … But during rush hours waittime may be higher