Not all of these will be entirely original, as many of these headcanons are agreed on by many fans (e.g. Jak has nightmares after being tortured), but any that are not my own and are very specific will have sources. Eventually they will be split into categories as well, perhaps each with their own separate page.
Daxter was homeless, perhaps orphaned or abandoned as a child. In The Precursor Legacy, Daxter is never shown to have a home, is accused of stealing, and is on the receiving end of harsh comments from Samos about his hygiene ("Looks like you finally took a much-needed bath!" he says, after Daxter fell into the pool of dark eco). Equally, there is no mention of his family, and the general attitude towards him seems to be one of disdain, most notably from the adults in Jak's life, Samos and the Explorer. These all point to Daxter being homeless, so naturally the adults around Jak would want to keep their golden boy away from the loud-mouthed, grubby urchin that likes to hang around.
As for the cause, seirei-bh on Tumblr theorises, based on Daxter's personality, that he was abandoned, since he always needs to be the centre of attention, as if he's afraid that people will leave him if he isn't entertaining or interesting enough, hence his lies and embellished stories.
Daxter may believe himself to be without family, but in fact, as of The Precursor Legacy, he has at least one living relative: the NPC, Boggy Billy. Jak and Daxter meet him in the Boggy Swamp where they have to help him lure his pet hip hog back after it ran away. Boggy Billy bears a striking resemblance to Daxter: pale skin, orange hair, buck teeth, and even a tendency to overindulge in alcohol (which can run in families). Personally, I don't think Boggy Billy is his father, but perhaps an uncle or cousin, someone Daxter would not have known well enough to recognise, but who was still close enough biologically to Daxter to bear such a resemblance.
Daxter likes to collect pebbles, pretty bottle caps, and other knick-knacks. There are two primary reasons for this: growing up homeless, he never had the opportunity to collect pretty things because he had nowhere to keep them, so he compensates as he gets older; otters also tend to have a favourite rock, and since Daxter now occupies a part-otter body, perhaps the trait stuck.