Is it a veggie or ornamental garden?
I wouldn't use bark chip on a veggie garden; aside from just deing a pain, it's almost imossible not to bury what is basically chunks of carbon amongst your plants, starving them of nitrogen.
In the veggie garden, I use thick as I can mulch of anything soft-ish. Pea straw mixed with grass clippings is good. I've never sheet-mulched a productive garden, but have done pretty much what you ask about for killing a lawn.
I used a mix of boxes and paper. If you can, boxes are much, much easier. Just make sure everything has major overlaps.
I don't compost my ornamental garden, I only grow really tough plants with a lot of fresh tree chip as mulch.
After realising I was being WAY too stingy with mulch, I basically don't weed.
I'm lucky though, my only pernicious weed is oxalis, so different rules may apply...
Not sure what you mean by "black tarp", but I avoid anything that costs lots of money and/or won't eventually break down.

as feijoas asked: veggie or ornamental?
As he stated I would leave the bark chips for the latter and I wouldn't use anything in either garden that wouldn't eventually breakdown and become part of the soil.
Mulch heavily and I'm sure you will be pleased with the results.

When used as mulch, layed on the soils surface and not mixed into the soil, wood chips will not "rob" the soil of Nitrogen, however, shredded leaves are a better mulch. The newspaper does not need to be wet prior to laying it in place, I find wet peper difficult to handle and it tends to tear apart. I would not use a tarp, or any other impervious type of material because those things tend to keep rain water from getting to the soil and the plants and they can also keep air from the soil.
Otherwise your plans look pretty good.

I thought it was gonna be about hens...