Upvote:7
Re. your first question, the somewhat traditional account seems to be that the Achaemenids indeed had a province/satrapy called Parthia (e.g. mentioned here and here). But the people who would later found the Parthian empire did not live in Parthia at that time yet! Rather, towards the end of the Seleucid empire that area was invaded from the north by Iranian steppe peoples (Dahae or Parni) and these invaders went on to found the ruling circles of the Arsacid or "Parthian" empire. So if this account is correct, it seems that the predecessors of the ruling class of the Parthian empire lived outside of Achaemenid control.
Re. your second question, there are several bilingual inscriptions from the third century (but apparently not later). There is also the observation that only two of the seven ancient houses of Iran are actually attested in Arsacid-era sources and the other five may just have made up their genealogies. Both of which IMHO implies that, at least for a while, being Parthians was not a disadvantage in the Sasanian empire.
Re. your third question, there seems to have been a post-Islamic regional dynasty in Northern Iran that traced its origin back to the House of Karen. If I understand correctly, Christensen in L'Iran sous les Sassanides, p.42 asserts that the language (or dialect) of the Parthians is related to those dialects spoken in northern and central Iran today.
This is all just googled and I am by no means a historian, so please take it with a grain of salt.