Attach partition wall to lath and plaster wall

I think you would be better off using a remodeling box (with the flanges that rotate out to grip the wall) and ignore the stud altogether. With a remodeling box, you just have to cut a hole big enough for the box, whereas if you use a regular box, you have to cut away more so you have room to nail it into the stud.

The main difficulties you'll face are that you don't know exactly where the lath is placed within the wall, and that the plaster will be brittle with age.

Start by using something sharp to gouge out a small hole in the center of the spot for your new receptacle; I use the pointed tip of a drywall saw. Once you've exposed the lath, expand the hole up and down until you find the edges of that piece of lath. The other lath are going to be approximately the same size and spaced up to maybe 1/4" apart. Adjust your planned spot up or down a little so that at least one edge of the box lines up with a gap in the lath -- it's easier to cut through the plaster on its own.

Expand the hole until it's big enough for the box, then use a jigsaw to cut away the lath. Vibration from the saw can cause more damage to the wall so take it very slow and easy. (Another reason to avoid the stud: if the lath ends at that stud, you'll only have about 1/2" of it to hold the plaster in place.)

Given that it's an exterior wall you've probably hit brick.

This would explain both why you've not found a stud - there aren't any, and why the nail only goes in 1/2" - 1" - you've hit the bricks.

Given that the plaster is crumbling, the simplest solution would be to drill a hole with a masonry drill and use a screw (or screw hook).

If you want to hang a lot of things on this wall then it might be a better idea to install some sort of rail - either a traditional picture rail or a more modern alternative and hang the pictures etc. off that.