I have the Fender Acoustasonic 15 amp and I'm wondering whether I can plug in any overdrive/distortion pedal so that when I'm playing my Fender AmSTD it sounds decent.
You can definitely use your distortion pedal to make a different sound. Whether or not that is better is going to be entirely down to what sounds you like yourself.
As ever with these sort of things, try it and see.
Yes, this works, but it will probably result in a rather annoying, tinny or “dentist drill” like sound.
Electric-guitar amps don't include any real HF speakers, only a single size of low/midrange woofer, which doesn't attempt to give a neutral, HiFi or studio-linear frequency response. The response curve has generally quite a mountainous shape in the midrange and really steep cutoff at high frequencies. For acoustic instruments this would be considered annoyingly uneven, but for electric guitar this shaping is “just right” to give the standard sounds. In particular, distortion effects tend to rely strongly on the HF cutoff: hard clipping embeds sharp transients with lots of HF content in the signal, but electric-guitar cabinets smear those out to just the right nice bright sustained lead sound.
An acoustic-guitar amp OTOH focuses on preserving transients as much as possible, to get the most direct, crispy response to the notes you're playing. That's great for acoustic guitar (duh), but it means that many distortion effects will come out way too treble-rich, in fact the result may seem either ruthless-synthetic (similar to a square-wave synthesizer) or just plain broken. With clever EQing you can somewhat counteract this – in fact there are some cool recordings with distortion without guitar cabinet, e.g. Jimmy Page used this sometimes.
But it's not really helpful if you're after a classic overdriven electric guitar sound. In that case, I'd recommend getting a small electric-guitar amp instead and cranking its gain to get some distortion.
As the good doctor says, yes. In fact, you can plug any guitar through any distortion (or any other) pedal into any amp and you'll get some sort of sound. Of course, only you will know if that sound is what you're looking for. There is but one way to find out Try! Be aware that the volume control on the guitar then sort of turns into a how much distortion do you want control.