Roots: Greek mythology (and possibly Hindu if Kalkin is a variant of Kalik), Lucie refers to the Doctor's "Miss Marple routine". The theories of Erich von Daniken ("they're coming for us - the Chariots of the Gods" - which are actually helicopters), Zeus refers to Kalkin as 'the Prodigal Son'. Red Riding Hood ("my, my, what big words you have" "all the better to know you with"). Lucie quotes Oliver Cromwell "warts and all". The Doctor quotes from Kipling's The Female of the Species. Lucie says "Its like Romeo and Juliet all over again" and sings the refrain from the Beatles' When I'm Sixty-four. Asimov's positronic brains (although see: Links)
Technobabble: Synaptical kinesis. Positronic circuits
Goofs: If Kalkin is only a couple of months shy of being too old for the RSK machine, why doesn't he just lay low with Sartari until he's of no use [possibly Sartari is younger, or perhaps the threat of death would remain due to their disobedience]
Fluffs: In scene one Kalkis seems to say "Juanita climb a little higher".
Dialogue Disasters: "Zeus is angry as hell."
Continuity: Remote Synaptic Devices (RSKs) are banned in every civilised world and only work between bodies which are genetically identical (although there are still limits - Ganymede's synaptic links have yet to cement themselves in his brain; thus, a mind-transfer before the age of 16 won't hold and Kalkin is almost too old for the procedure). Clones are fostered until they are of use or reach an adult age to serve the gods as a vocation). The RSK used here is an antique ten times over, according to the Doctor. Use of RSKs nevertheless is not an exact science - such complications with the procedure can involve memory loss between generations and mis-transfer if the subject mind is not a perfect match.
Caleva may be the name of a country or the world where it is situated. Of the thousand-year old Erath colony on it, the entire population has descended or been cloned from the original crew. Past (and since deceased) 'gods' included Athena and Posedon (nee Geoffrey). It is not explained why the civilisation is based (loosely) on that of ancient Greek gods, or how the population originally fell for such a fanciful ruse. Zeus is, by his own admission, over a thousand years old.
Future History: Helicopters and guns are still in use (although the planet's technology may not have progressed for several centuries)
Links: The Horror of Glam Rock ("Lucie of the M62") the idea of a positronic brain being at least partly organic is postulated (or misappropriated) in Power of the Daleks.
Location: Caleva, possibly circa 34th century (assuming the colony dates back to a thousand years past the earliest Earth expansion).
The Bottom line: 'What good is 'forever' without her?'
Once again the Doctor is invoked as a Greek god, while the world outside Zeus' palace is barely given description - indeed the scenario of a backwards planet with seemingly immortal rulers exploiting technology to rule over a forcibly ignorant population is more than reminiscent of State of Decay.