Languages constantly change, so it is actually not appropriate to call one language older than another. Attestations of a language, that is, written records that have been preserved, can be dated, but they simply reflect a language frozen at a point in time.

Old Tamil, which is not the same as modern Tamil, has attestations as early as the 2nd century BCE. Old and modern Tamil belong to the Dravidian language family, a group of related languages which all descend from a common ancestor, Proto-Dravidian. Other Dravidian languages include Brahui, Kannada, Malayam, and Telugu. There are some common misconceptions around because there is influence both on the Dravidian languages by Sanskrit and other Indo-Aryan languages and the other way around that they are related, but it is widely accepted by the scholarly community that this is contact, rather than a genetic relationship (meaning they don't come from a common source).

There have been attempts to link Dravidian and Elamite, starting with McAlpin 1975, but these are not accepted by the wider scholarly community. Elamite is a language isolate (meaning it has no known genetic relationships) that was spoken in what is modern day Iran, and is attested from around 2,000 to 500 BCE. There have been other attempts to show a relationship between Dravidian and other language families, but these too have not received widespread support.

With such a question, we must keep in mind that things like "classical language," and even the divisions between different languages, are not absolutely defined. Classic Tamil, though, is generally dated no earlier than ~250 BC, while Both Classical Greek and Chinese can be solidly placed in the 5th century BC.

As for the second question, Tamil, like other languages in southern India, is a Dravidian language and not connected with the Indo-European languages to the north. The Sumerian period was 2000 years prior to Classic Tamil; the Sumerian language itself is considered an isolate with no known relations. Thus, there's not many ways to connect it. Unfortunately, research on Dravidian languages is limited, so we haven't been able to track it that far back yet.