Dr. Vezza couldn't have been 37 in 2002. He claimed that he was conceived during the great Northeast Blackout of 1965. However, the blackout occurred in November of that year, meaning that he would have been born in 1966, and therefore making him 36 in 2002 unless he was mistaken about the circumstances of his conception.
After Monk and the others first meet Dale the Whale, they go see a judge. Monk is placing jelly beans on a scale. When the scene starts there are 4 beans. Monk places 6 more beans and when he is placing the 7th making a total of 11 the close-up shows only 6 jelly beans.
In Dr. Vezza's office helium balloons are shown fully upright. The doctor says they are from his birthday a week earlier. Helium atoms are thin enough to pass through any material used for balloons, therefore will not remain afloat for more than a few days.