Do δικηγόρος and δικαστής have some background in common
Yes. δίκη is "right, justice, judgment".
From that comes δικάζω "to judge", and a δικαστής is a "judger", someone who judges (we say "judge" in English).
And a δικηγόρος is a "justice-speaker": an advocate or a lawyer. (The second part is related to αγορά "agora" and originally meant something like "to speak in a public place".)
Is "jurors" δικαστάς?
Not in modern Greek.
Ancient Gk "jurors" was δικαστάς
At least in the accusative case.
Have a look at https://en.wiktionary.org/wiki/%CE%B4%CE%B9%CE%BA%CE%B1%CF%83%CF%84%CE%AE%CF%82 and compare the declension tables for "Ancient Greek" and "Greek".