The entire Indian opposition to the LTTE is based primarily on your point no 1 followed by their war on the IPKF.
As far as their killing Tamils go , there were more than a dozen Eelam groups operating out of TN & SL in the 80's most of whom were exterminated by the LTTE. That didn't prevent Indian intelligence & the state supporting them although they weren't too comfortable with this internecine warfare & LTTE's policy .
Point 2 & 4 needs to be substantiated for they're baseless in my opinion. The SL Tamil diaspora in the west were extorted coz they had kin back home not the Indian Tamil diaspora whom the LTTE neither targeted nor could they extort. Having said that there were expatriate Tamils of Indian origin who did extend support to the LTTE in spite of the Indian state's opposition to it & it's being proscribed post RG's assassination.
This ban is the reason why action was undertaken against the LTTE apart from SL prompting I suppose , which is why I'm suggesting we relax our vigil.
Perhaps the GoI is playing the waiting game to see how far is SL willing to withstand Chinese pressure.