Can I use yeast from a bottle of beer?

Yes, yeast can be recovered from bottles of "live beer". IIRC, Cooper's is an example of an Australian brand that has live yeast in the bottles.

However beware as some companies micro-filter the beer to remove the yeast used to brew and then bottle the beer using a high flocculation yeast to improve beer clarity on consumption. So it is possible that the yeast from the bottle, while being active, is no the strain of yeast used to brew the beer.

It is also worth noting that the yeast recovered from a bottle has to be grown further in a starter culture (with oxygen/air and nutrient available) as there will not be enough yeast present to correctly initiate fermentation. It is not usually possible to empty a live beer into a wort and find it will ferment "optimally".

If all else fails and the need is great - use baking yeast or even a wine yeast. I often use a champagne yeast to ferment a light, dry beer.

First of all, can you please elaborate on the "no action" observation: how did you conclude there was no action in the fermenter? Just "no bubbles in the airlock" doesn't count, as your fermenter may be not completely airtight. Best of all, you should have checked the original gravity of the wort, so you could compare it to the current and see if it decreased. If you didn't do so, check if there is a foam head (krauzen), and check whether the wort became murky from wort. There may be some action.

If none of that is the case, and your wort is completely dead.. Tbh your best course of action is to find some (any) ready-to-use brewing yeast. If I didn't have any brewing yeast around, I'd probably pick distiller's yeast (first choice) or even baking yeast (second choice).

The reason is.. the wort is not 100% sterile, at best it's sanitized. Which means there is some level of random household bacteriae/fungi from the beginning. Should you have pitched a healthy dose of yeast, that yeast would have eaten all sugars, leaving nothing to "weeds". Making a healthy dose of yeast from a bottle is possible, but it's not a 1 day job (and perhaps not even a 3 days job): there's very little yeast in a bottle, it's very dormant, you need to grow it using step starters -- and all this time your wort will sit there, increasing a chance for developing an infection.

While I was writing the above paragraph, I thought of a way to buy yourself a bit of time. You can re-boil your wort briefly and pour it into a food-grade temperature-stable airtight container, ideally leaving no air at all, screw the lid, turn it upside down, to get the lid sterilised by the hot wort. Well, you need to have such a container in first place. Here I'm describing briefly the "no chill" technique popular among homebrewers in Australia. It effectively makes "wort preserve". Then you can shop for yeast or play with growing your own from a bottle. The downside is that such preservation increases chance of Maillard reactions, so your beer will become a bit darker and more oxidised (but hey, it's still gonna be beer).

Yes, this is a common method of getting yeast for a particular style of beer. A few things to note:

1. There's not much yeast in a bottle - so most people use a starter to grow up a larger culture. However since your beer is already ready, I'd suggest instead getting a few bottles to increase the pitch (I'm afraid you'll have to drink the 'waste' beer!)
2. Try to choose beers with a long use-by date to get more live yeast - and avoid ones that have been left in warm, sunny places in the shop. Most people recommend sterilising the top of the bottle when opening/pouring it to ensure no bad bugs get picked up.
3. Pick a style of beer close to your beer - so don't use a stout for a lager, or a saison for an IPA (unless you want to mix things up!). Also be aware that if you do aim for a particular style, some breweries use a different yeast for bottling than for brewing, so you might get a high carbonation, low flavour, high flocculation yeast if you're unlucky (Though it'll still be beer!).

Good luck!