Yes, that is where it comes from and you might find this post I just wrote to someone else interesting. "The Greek συ,συμ,συν are often equivalent to the English co,com,con and there are many examples: concert->συναυλία, connected-> συνδεδεμένος, companion->σύντροφος, etc. " And thus "contemporary" ->"σύγχρονος". "Συ" meaning "together" and "χρονος" meaning "time".
Yes, as with English it means "occurring at the same time". And again as with English, it is often used as "modern" but it could be any time period. In English, we have "con" (with, together) + 'temporary' (time). A corresponding configuration exists in Greek. The prefix "συ/συν/συμ" the same as the Eng. "con" + "χρόνος" meaning "time".
Just to comment that "modern day" does mean modern age in the right context. However I think it would only be used adjectively (so would not be correct here), and refers to something associated with the past being recreated in some sense now, eg "Burgundy is in modern day France", "he's a modern day Beethoven", "modern-day slavery" etc ...