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Some tree leaves tend to get matted down when wet which inhibits air and water penetration, things that are necessary for aerobic decomposition. This is especially true of large tree leaves. So it is best to run these over with a mower before using as a garden mulch. Another option is to intermix straw/small sticks/pine needles with leaves to prevent matting. Shrub leaves are generally small enough that shredding isn't necessary. When mulching don't put any up against plant stems. All leaves make a good addition to the compost pile. If added to the compost pile, they don't need to be chopped first, although chopping/shredding speeds decomposition. It may be necessary to fluff the pile to correct any leaf matting.

Yes or be prepared to use herbicide weed killer.

If its only rock ground cover in the area then you can use plastic. But if you have plants among the rock that need watering you should consider a landscape weed control fabric.

https://www.gardeningknowhow.com/plant-problems/weeds/weed-barriers-in-gardens.htm

Yes on using the wood chips as mulch.

No on planting in the raw wood chips/sawdust material. If you want to plant in that spot, dig out the backfilled sawdust and replace with soil and compost mix.

I think that you are doing the right thing already. It could be likely that the weeds are coming from seeds blown in or tracked in by animals. You can lightly spray them with vinegar, and without spraying the ground much at all. This will kill the weeds quickly. Just be careful not to get the vinegar on your plants that you like.

You do not need to remove the old mulch. Add a bit more on top where there are bare spots. The mulch slowly turns into compost.

To use the wood as a soil amendment, the wood should be completely composted, with a dark, crumbly and uniform texture with no recognizable pieces of wood or other materials. Fresh wood mulch ties up nitrogen as it starts to decompose, although it will eventually become a source of nitrogen itself, so it should be aged before applying it around shallow-rooted, young plants.

https://www.gardeningknowhow.com/garden-how-to/mulch/wood-chip-mulch-information.htm