Why are the people of Troy called Trojans?

[Dan Diffendale]

Okay I've done a search to see if anyone has asked this seemingly basic question before but haven't found anything so I ask you, why in the world are the people of Troy not called Troyans? Where did the 'j' come from?

The culprit is our old friend the Latin language. English "Trojan" comes from Latin troianus, that is, someone from Troia, "Troy." I can't tell you the first attestation of "Trojan" in English (anyone have access to an OED?), other than that it comes in during the Middle English period.
Already in late Latin the semi-vowel i was being pronounced /d3/ (~dj), so the answer in this case is that the sound was borrowed along with the word. For an analogy, think Latin Traianus, English "Trajan."
Troia lost its Latin ending in English, and the i no longer functioned as a semi-vowel. So we say Troy.

I'm still trying to figure out why the Romans and we ourselves call the Greeks "Greeks"

The Graikoi were apparently originally a Boeotian group who colonised in Italy. From their name the natives generalised to all the Hellenes, who became "Graeci" in Latin. English then took it from Latin.