The "bars over hub" rule is a general guideline. Sort of like the "stand over" test it helps give the salesperson a general baseline as far as what size bike the customer should begin test riding.

As far as stems being too short: stem length does, to a degree affect handling. My (strictly personal) preference on a road bike is a stem 100-120 mm long. I've seen shorter and I've seen longer, but I ride what feels right to me (110 on my current bike).

Your elbows shoud be "relaxed" . If you elbows are "locked" you are tensed up and this can cause shoulder and neck pain. Slightly flexed elbows also absorb some of the road shock. You have the ability to control the amout of flex. Try this on the trainer: Sit on your saddle, perched on the sitbones. Now lean forward, place your hands on the hoods and bend your elbows. If you cannot maintain this position you may simply not yet have the strength/flexibilty OR the bike may indeed be too long in the top tube.

If the bike is too long...yes you will have to switch stems. It's not a perfect fix, but it's cheaper than a new bike.

You may also benefit from different handlebars. The has been good discussion on this on this board in the past (try a search in "general topics" and in the gear section).

Good luck.