If I successfully disable "Whatsapp web is currently active" on the notification panel, are the individual and group messages and alert silent too?
In case you have disabled the notification by blocking it, you've probably blocked the whole WhatsApp from showing any notification. In that case, you'll not get any notification from WhatsApp.
However, in my phone, it doesn't show any such notification at all when using the web feature. Even I've seen many others use it, but no notification is shown. That's what is making me doubt that you have the correct WhatsApp. Maybe I'm wrong, and it doesn't show the notification in India, but that shouldn't be the case. Please make sure that this is the WhatsApp you've installed, because there are some more similar looking apps in Google Play.