I haven't really noticed this to be true at all. On your point of promotions, I'm thinking about both my current department chair and the one for the department I'm considering a move into...both are on the short side for men. It sure hasn't affected their ability to be promoted. Our interim dean is a rather short guy too. (Several of the senior faculty I teach with are also short, and we joke about it that we're "short" staffed in the course we teach together
). Of course, if you sit around moping because you're short instead of getting things done, you're not going to get promoted, but it has nothing to do with height.
It's also a rather broad generalization to think women all want men 5'10" and taller. Plenty of women are much shorter than that and are not particularly attracted to guys who tower over them (I'm sure not...my current boyfriend is about 5'4" or 5'5", and that is absolutely perfect for me...by the way, his height isn't hindering his promotion or earning power either).
I actually think it's about the same, but attitude can get people to overlook your height. It wasn't until I stood next to another faculty member who is also short that someone noticed we were close to the same height and that meant I was short too. Even though we were about the same height, nobody perceived me as short, but they did perceive her as short (they call her the "pixie" of the department). The funny thing is that most of the time, she wears heels, so even looks taller than me (I mostly wear flats), yet they noticed she was short but not me. It comes down to attitude, and sometimes posture.
Oh, good grief, I don't know who gave you that advice, but it's WAY off the mark! Being a bigger jerk just makes you a bigger jerk. If you're a jerk, then you won't get promoted because of that. If you try to "compensate" for height, people will pick up on the attitude problems and view you negatively. Just act like anyone else...be confident, work hard, work smart, and you'll earn respect the way everyone else does. Learn to be assertive without being aggressive (this is something women used to have to deal with too...for some reason, it used to be that women were given bad advice about needing to prove themselves in a way that made them come across as b****es rather than a strong leader...once taught better how to be assertive without being a b****, advancements and promotions come along a lot faster).
Most of the short guys I know are NOT weak at all. Odd they'd assume that, unless you're also very slender (and then it's not the height, but the slender build that leads people to assume you're weak...I know some wiry guys who are over 6' tall who run into the same problem of people assuming they're weak because of their build).