Is there a brand and model of the mixer in mind or is this just a general question. For specific mixers this sort of information enables people to help much more so. Block diagram and manuals can be downloaded and checked.
Generally though there are probably enough outputs to feed your main speakers for the PA, a set of other outputs of some description to drive studio
(active I assume) monitors (e.g. Control Room Out) and the foldback sends driving your foldback system. All without patching. And there will be separate master controls feeding each of these scenarios. It would be a simple matter to adjust the send level to the required outputs.
A lot of decent PA/studio mixers would have built in features that would be similar to what the Dangerous Music Source can do. That is good for connecting to a DAW to give all that input output monitoring connectivity for a computer plus a variety of other input sources.
A powerful combination if you can afford it would be the Dangerous box and a nice live mixer feeding a PA and monitors. Then the computer could be patched directly into the PA mixer. Otherwise your interface attached to your computer will work fine as it is next to the PA mixer.
I have often done multitrack recordings and operated a PA at the same time. What works real well there though is to use a fast interface on your computer. All the input sources go to them first then into the software. With a good interface extreme low latency is possible and it is fine to have the PA being fed from special buses within the software. Without any perceivable delay. More interesting PA monitor mixes can bet set up that way. But sources can also go direct to the mixer and use the direct outs from the mixer to feed the DAW as per the more often conventional way.
I would be looking for the PA mixer have the capability of sending separate channels digitally to your DAW as well. This will make multi tracking a snap too. There are some e.g. Presonus and these all work as a full PA mixer and with Studio One capture software making it simple and easy to capture everything live.