Aero bars and road bike seat tube angles

You can put aero bars on a road bike, but you have to be careful about positioning as it will affect handling (road bikes are generally not designed with the expectation that the rider is going to move the center of gravity that far forward).  The seat tube angle is only one part of that and, actually, you can ride aerobars "steep" or "shallow".

You probably get passed downhill because aerodynamics become more important as speed increases.  You are presenting more drag than they are.  It can be more difficult to climb aggressively in a tri bike.  But generally if your bike fits and you are adapted to it and you have enough gears, you can climb just as well.

On a road bike, you may be better off with "shorties", basically a shorter aerobar that won't put your weight as far out towards the front wheel where handling may become more of an issue.  Other than that (and without access to a windtunnel to determine the best aerodynamic position for you), it's mostly personal choice.

The best thing to do is probably to pay for a professional fitting and have them set up the bike with you.

No.  the angle is different (i.e., not as steep or "forward" but many many many people put aero bars on road bikes and do quite well. 

I'd venture it's got nothing to do with aero bars since, after a certain grade, you shouldn't be climbing in the aero position.  If you're passing someone that is climbing an 8% grade in aero, then yeah, it's probably due to aero bars.  Maybe they're heavier than you.  Maybe you're a better climber and chicken out on the descent   Too many factors.

First, what's your price range.  Second, how much does weight matter.  Then fit is also important - how comfortable is it?  S bars?  etc etc etc.

Go to your local bike shop and get a proper fitting