I see in the 2nd photo you have some chain link, if you're planning on using that in your run fencing you can simply jam a branch through the chain link holes and it holds up pretty well. I have 2 roosts in my run that are simply branches stuck in the corner in that fashion.
If you're not using the chain link, you can try something as simple as wiring the ends of the branch and then looping the wire through the wire fencing you have, maybe around some of the wood too for extra support.
That said, I've never had to remove the branches for cleaning, the birds poop off the side, never on the branches. The rain and wind cleans up any other dirt.
Or... if you have a lot of branches to pick from, sometimes I find really large, twisted branches that can be placed lying directly on the ground, so it's a free standing roost. I have one in the back of my run that the birds can climb up on one side and down on the other, or they can fly directly up to the higher part in the middle.
As far as height it depends on your birds and also if the run is covered or not (you don't want them to use the branches as a stepping stone out). I have mine fairly low, probably about 18" and 24"... maybe 24" and 30", I'm not completely sure. My birds enjoy it a lot.