Does anything determine their personality traits? Or is it random?
Also in the park in the morning a blue guy offers me a job or something, if my tama does it long enough will she make money?
The personality of your tama depends on what items they play with/eat
And for the jobs your get 250 gotchi points an hour and if you do the same job for 30 hours without doing any other job in between you get a special item... supposedly.