Sometimes shortening the reach will not help. If you are seated way too forward you are putting more weight on your hands. This is what I experienced. I moved my saddle back and I was able to bend and reach further. Keep in mind that pedal strokes comes first when fitting into a bike.
we cant address fitting problems with out seeing you ride under load so the only advice you get from this thread that will truly help you is to go get fitted for the bike.
I hate when people say check out this calculator it will fit you. if it was that easy companies would produce the same geometries for all frames. it just doesnt work that way. we are all different you may need a shorter stem you might need to relax your elbows you might need to raise your seat all of which will change the dynamics of your fitting
lower back pain is usualy an indicator that your bike is too short or that your seat is too low
please put your bike in a trainer and ride it in front of your computer so i can see you ride under load
As far as fit calculators go, I have a high regard for the one at Competitive Cyclist for getting you in the ballpark. Most fitters start with a similar calculation system and then work on the individual adaptations. Seeing that you're already in pain, though, I'd recommend a session with a live fitter if at all possible.
Without seeing you on the bike, I can only suggest what helped me (and this was after I bought a $3K carbon "plush" bike with upright geometry to address the problem) - core exercises. Bicycling mag has published many great articles about this, some long, some short (e.g. "fit chick"). The latest is a really good one - March 2009, page 54.
Trek 1000, two mtbs and working on a fixie for commuting.
The only time I had elbow problems was from the insane nervous grip on the bars when I was a new rider - learn to relax and see if that helps you elbow pain. Other than that, competitivecyclist.com's fit calculator is a good start and perhaps you can take pics of yourself on your bike and post them up, probably better in the road forum.
There are a lot of variables, and if you have no idea about proper road bike setup, then you should see somebody who does. The only other solution is to start by fitting yourself using the tried and true, traditional bike adjustment methods and then ride. Such a position might need to be fine tuned, and it may not provide you with the absolute most power that you can produce so you can win the big tours and get the big endorsements, but it won't be far off what you need for normal riding. Then, if you still have pain, you have a base point to move away from in terms of saddle and handlebar positions. The old school fitting rules of thumb may not be absolute "rules", but they are a reasonable starting point, and when you later move things away from that, you know where you are going with it.
A higher and/or closer handlebar is not automatically be the solution. Maybe your saddle is too far back, maybe it's too high, maybe it's too low, maybe your reach to the handlebar is too short or too long. Who knows without seeing you riding. A still photo is of limited use. This is why a good bike fitting usually usually mounts your bike on a trainer of some kind.
The low handlebar position is a bunch of extreme bull, and the high handlebar position is a bunch of extreme bull. Neither is your solution. You need a reasonable, well-balanced, moderate road riding position.