For example, we can't go back much further than 500 years in English and still be able to fully communicate in modern English. Are there languages whose modern speakers can fully communicate with speakers much longer ago, like thousands of years back?
Basically, no.
One shouldn't confuse written language and spoken language. Some languages are very conservative with their writing systems. English preserves its orthography from before the Great Vowel Shift, written Icelandic hasn't changed much since it became written, Tibetan writing still uses orthography from over a thousand years ago, and Chinese writing, because of its nature, hasn't changed that much in over two milennia (discounting the character simplification in China).
However those are all writing systems. Spoken language evolves continuously, whether it's written down or not, and writing doesn't influence its development all that much. For example, Tibetan has split into several varieties, such as Lhasa Tibetan and Amdo Tibetan, that are mutually unintelligible in spoken form, but use the same written standard. Likewise, in Hong Kong, the written standard uses Mandarin grammar, and thus is more or less intelligible to Mandarin speakers, but spoken Cantonese is not (nor is written Cantonese, that is, Cantonese actually written down the way it's spoken).
There are many other examples, but the main point is, even if the writing is very similar, the spoken forms may vary greatly. This is doubly true of languages where the written form is very conservative and thus very dissimilar to the spoken form. Languages change all the time, but some languages don't reflect these changes in their written form (or can't, in the case of Chinese).
The discipline that deals with how languages change over time is called Historical Linguistics. You can find a lot of general information in the following books. You're also welcome to ask more specific questions at /r/linguistics.
Campbell, Lyle. 1999. Historical Linguistics: An Introduction. 1st MIT Press ed. Cambridge, Mass: MIT Press.
Joseph, Brian D., and Richard D. Janda (eds.) 2003. The Handbook of Historical Linguistics. Blackwell handbooks in linguistics. Malden, MA: Blackwell Pub.
Millar, Robert McColl, and R. L. Trask (eds.) 2015. Trask’s Historical Linguistics. 3rd Edition. New York: Routledge.