Tri Bike vs Road bike/aero bike questions

You can add aero bars to your Domane...but it won't be ideal.  It will still "work," but just not nearly as good as a true tri bike, and not quite as good as it may on a more aggressive road bike such as the Madone.  The reason is that the stack (how high the headtube/handlebars are) of the bike is made to be higher on the Domane, thus adding aerobars won't get you as low or as aero.  It will still be more aero than riding in the traditional position with your hands on the hoods though.

If you can get a pair of cheap adjustable aerobars, I would add them.  If nothing else, it's an additional position you can ride in.  If you're a MOP/BOP rider, then you'll be out on the course for 3+ hours and it's nice to be able to take some of the pressure off your hands by riding on the aero bars for periods of time.  Plus you'll be faster when doing so.

But one thing I would advise is that you do not change your road bike position when adding the aero bars.  Slap on the aerobars and adjust them as best you can without touching anything else.  If this means you can only ride in the aero bars for 5-10 minutes at a time...so be it.  It's still better than nothing.  Don't worry about the weight penalty of adding aero bars either.  The difference is likely very small and can easily be made up by riding aero for a short time.

Exactly.

Some of the best-summarized information I've read out there is here in Choose Your Bike:

http://www.elitebicycles.com/tech_tut_fit.php

I'm going to agree with the bike shop guys. If you bought the Domane because it felt "more relaxed" then chances are putting aerobars on it and doing nothing else to the bike (not re-fitting) would just feel awkward.  You'd just bend at the waist more and that would just close your hip angles and reduce your comfort and power. Go for a ride and spend 20 minutes on the drops (not on the hoods, not on the tops, but down low).  That is close to what riding an aerobar will feel like.

But if you want to try them, I suggest the lightest possible ones, like carbon or aluminum. Too much weight on the front-end can make the bike very skittish. No big old-school hoop ones, instead separate bars.  The "ski jump" ends are usually more comfortable than the S bend (also known as R bend for "racing bend), and ignore straight bars.  Profile Design makes good ones. 

The purpose of a tri bike is to put the huge non-aerodynamic rider in the best possible low-air-drag position known as "aero position".  The bike has significant changes to the geometry of the entire bike to accommodate this. I'm going to link some images off the Internet from http://www.bikesportmichigan.com/bikes/difference.shtml, you should read the article.

And here's another image showing real bikes and how the tri bike rotates the entire rider so the hip angle in aero is still about the same as the road bike rider.  Tri bike aero position isn't just bending at the waist.  To answer your question about sizing, you can't really compare road to tri or even different makes.  People measure from difference positions.  There is a system of measurement known as "stack and reach" which is more standardized. Find that on the Internet too.

Here's an image that's floated around a while (from another article http://gearwestbike.com/articles/tri-bike-or-road-bike-pg54.htm)

 Pretty clearly shows the differences.

I'm with you (and the bike shop guys).  The Domane has such slack and relaxed geometry that slapping on some aerobars would probably cause more harm than good.  On a Madone... okay.  Not on a Domane.