I'm not going to try this out until I'm completely sure, so here's the deal.

I have a WA15G amp. I'm told it's not the best thing ever for playing live, being a 15-watt and whatnot (I'm currently saving up to buy a better amp) but it works pretty okay. Mic'ing it up hasn't been mandatory so far because the places I play are small enough that people can hear it directly, but the next place is going to need either mic'ing it up (most likely I'll use this) or extra oomph another way.

Sound guys get a bit offended when I don't plug my stuff into their board tbh, but the one time I plugged my guitar directly into the soundboard at his insistence, it sounded terrible and we went with the amp. That's out of the question here anyway, since I don't have a distortion pedal yet (yeah, I know) and need to use the overdrive channel on the amp.

Aside from mic'ing it up, obviously, I'm wondering if the phones jack on a combo amp can be used to plug it into an external PA system, soundboard first of course.

Can it? I'm not gonna try until I'm sure, too many horror stories. Sorry for the vague info, it's all I have.
By "external speaker jack" that sounds much more like a power amp out which is not something you want to be plugging into a PA system. But a headphone out likely will be able to be used by a PA system. You may need a direct box or pad attenuation to knock down the level a bit as a headphone out signal is quite a bit stronger than a guitar or even mic signal. However, I doubt the headphone out will sound good at all. If your techs are smart they may be able to make it work.

The reason why guitars sound awful direct into a PA system (with nothing in between) is because so much of the guitar sound is done through the amp/speaker. You have to have at least preamp and speaker simulation to have a direct rig that can be taken seriously. And before you buy a distortion pedal, I would invest in a better amp before anything else.
Basically the headphone out jack just reduces the power to a level your headphones can handle and deletes the amps speaker, In the case of a PA or powered monitor or whatever , they require a line level signal or power if you will, Which is much lower than what's at your headphone jack, In which case you'd be hitting the PA with a hot signal. Bad things can and usually happen in this case. However as Will points out you can use a direct box or attenuator to tame the signal but depending on several factors results can vary, Actually I'm surprised your sound guys don't already have a DI box, But then again you are using a seriously cheap bedroom practice amp to gig with, So maybe not all that surprised.
So this is the amp? https://reverb.com/item/2921376-washburn-wa15g-guitar-amp-combo-15-watt-practice-amp
Yes, you can hand him this output, and though that would be hotter than a line level, you can turn it down and he can adjust his gain for the channel until it's suitable for use. As far as the speaker emulation... Well, just look at the amp. No offense, but the speaker is not really even good, so whatever the PA presents will be in the same wheelhouse. Plugging an instrument directly into the board is never a good idea. In the studio it works, because they take the signal and run it into various amps. Then the guitarist can get one good take and the engineer can re-amp the signal until they find an amp/speaker combination that sounds good. This is done via amp emulation software these days.

To make a long story short, yes, you can do this. When you want the true output of an amp, use mics, but when that's not an option, line out, then you can use phone out.