Is “knife-in-one’s-teeth (woman)” frequently used English? Can we use it for a man as well?

Having a "knife in one's teeth" is not a common phrase, although it is a common TV Trope. As the page for cutlass between one's teeth writes,

Taking disregard for the old saying "Don't run with scissors" to a whole new level, the Cutlass Between The Teeth is the tendency for usually Badass characters to run around with the handle (or blade, if they're feeling really tough) of a sword or knife clenched between their teeth.

It brings to mind the picture of a pirate, or someone who will do even dangerous things to achieve their goal. They may be gung-ho, and very take-charge--as @Robusto puts it, in short they are ready for action.

This imagery is in contrast with the "other" cherry-picking which was picayune or "trivial". It can be applied to either women or men. Neither phrase is frequently used in words, but the imagery does tend to evoke a pirate or someone similar.

The phrase 'knife in [someone's] teeth' is not a set phrase but is metaphorical for 'ready for action' based on the image of a pirate ready to swing over to another ship by a rope with his weapon in his teeth.

Despite the imagery, it is not gender specific, even though it would be more likely to be applied to a man (knife-wielding action). That might give the use here a little extra zing.

As an aside, many of Maureen Dowd's articles are a bit creative language-wise and depend on cultural references to fill in the gaps. So many metaphorical things she says are one-off's, obvious to a regular reader but never to be heard again.

No, it is not a common phrase.

However, it is a very common image in action movies, where the hero (or badass villan) will often put a knife in their teeth so they can perform some other activity that requires both hands (like swimming), and be sure to still have the knife handy. So its pretty easy to picture what the author is getting at.

Typically the knife-in-the-teeth badass person in the action movie is a guy, but there's no reason why it has to be.