What Happens When India Stops Hitting Asia
You are correct in that, sooner or later (and nobody can say just when) the Indian plate will stop crashing into the Eurasian plate, and the the two plates will effectively fuse into a single plate. The underlying mantle convection is likely to continue for a few million years, tending to displace the combined plate further northward, probably with a component of anti-clockwise rotation, which might result in a few micro-continental fragments breaking off. The Yakutsk peninsular is one such possible candidate. 

Probably the single most important result will be the fate of the Himalayan chain. Again, the underlying mantle is likely to take a few million years to gain litho-static equilibrium, so seismic rumblings will continue, but probably with much less intense earthquakes than we have seen recently. Fluvial erosion is probably faster than you think. Within 20 million years, and possibly even in 10 million years, all of the great mountains will be worn down to a pale shadow of their former state. More and more rivers will break through from Tibet, and drain southwards. The Himalayas will become much less of a climatic barrier than they are now, such that the monsoonal air flow, and much of the moisture than it carries, will spill over into Central Asia. The main rivers that feed into the Ganges will carry less water. The climate of Central Asia will change dramatically, but it's difficult to predict exactly how. Currently the air flow from southern Europe is deflected southeasterly across Iran and the Gulf, being effectively blocked by the Central Asian mountains. As these mountains are eroded this airflow will become progressively less hindered, making west central Asia wetter, and the Arabian Gulf even drier than it is already.

Glaciers and snow topped mountains are already in rapid decline because of global warming. The ice will certainly be long-gone by the time the tectonic conveyor belt grinds to a halt. 

The eroded debris will drain into the Bay of Bengal. The current effects of rising sea level will flood most of Bangladesh in the short term, but it will ultimately be reformed as erosion overtakes rising sea level. 

As for Tibet, one would expect a long period of erosion, mainly by deeply incised rivers dissecting the plateau, much as the Colorado River is cutting away at the Colorado plateau in North America; a process greatly aided by much increased rainfall.