.... If I refer to her as моя подруга, would that send the wrong message?
Yes, it most certainly would. A much safer way would be to refer to her as "моя знакомая".
I guess my question is really this: can women be друзья to men?
Yes, they can, but they usually don't refer to each other as "друзья".
If a man has a подругу, does that automatically connote a deeper and/or more romantic relationship?
Yes, it does.
If two women are подруги to each other, then they're just friends, right?
Right.
Would a woman refer to another woman as a друг (assuming it's just a friendly relationship)?
No, she would refer to her female friend as "подруга".