The Tamil-Japonic connection isn’t quite as off the wall as one might think at first glance. There’s apparently a strong Andaman-Indonesian language connection. The convention of repeat plurals seems to have found its way to Japan. There’s also some similarity between the Finno-Ugric languages, which are Uralic outliers in a sea of Indo-European languages, and Dravidian languages that have a remnant in Pakistan. Contact between proto-Dravidian-Uralic and Altaic languages is a real possibility.
If Uralic is close to anything, it is close to Altaic and Indo-European and probably even closer to Chukto-Kamchatkan, Eskimo-Aleut, Yukaghir and Nivkhi. Yukaghir may actually be Uralic itself, or maybe the family is called “Uralic-Yukaghir.”
There is no connection between Austronesian (Indonesian) and the Andaman Islanders. Austronesian is indeed related to Thai though (Austro-Tai); in my opinion, this has been proven. If the Andaman languages are related to anything at all, they may be related to some Papuan languages and an isolate in Nepal called Nihali. A good case can be made connecting Nihali with some of the Papuan languages.
Typology is not that great of way to classify. Typology is areal and it spreads via convergence. What you are looking in search genetic relationship among languages more more than anything else is morphology. After that, a nice set of cognates.
There is probably no connection between Dravidian and Uralic in particular. Dravidian is outside of most everything in Eurasia. It if is close to anything, it might be close to Afro-Asiatic. There also looks to be a connection with Elamite.
Dravidian and Afro-Asiatic are probably older than the rest of the Eurasian languages, and they were located further to the south. Afro-Asiatic is very old, probably ~15,000 YBP.