Plumbing – Peculiar external spigot replacement needed

Good for you for replacing this crappy old polybutylene pipe system. There's no reason at all why you can't do what you're thinking of. One thing to keep in mind is that since PEX is just plastic, it would easily be penetrated by nails or screws driven through the drywall in the future, so you want to make sure to run the new lines at least 2" away from any interior wall space that people might drive nails or screws into. For any places where you have to run it closer to the surface, you need to nail a steel shield plate into the part of the stud that the pipe goes through.

Toilets are typically supplied using 1/2" pipe, while it's common to use 3/4" to supply a spigot. You can certainly tee off the toilet supply and feed a spigot, though you might not quite get the flow you're expecting. Should be fine for watering flowers, vegetables, etc. Though may come up short if you're trying to water the lawn with a sprinkler. You'll also probably notice a pressure drop off, if somebody flushes the toilet.

If you end up supplying a spigot (however you do it), I'd recommend installing a new anti-siphon frost proof sillcock. The one that's there looks like it's seen better days.