The venting is wrong only because of the 2 90's right before the tee (when it goes up and over into the tee). The 3" doesn't help ofcourse but it may not be bad as you think. A larger vent is not always better! But I just don't like 3" regardless and it's now code (up here in BC) that all appliances that are natural draft appliances should be vented through a min. of 4". But you are still allowed to use 3" if you're doing a refit.
The vent riser (part coming off the top of the HWT) needs to rise up higher so that you have a straight shot over into the tee. Ofcourse the horizontal piece running over to the tee should be graded as to allow the hot products of combustion to rise.
Sometimes it takes up to 5 mins for a vent to heat up. Until it does you aren't going to have proper venting and may have some flu gas spillage.