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Yes. Woodchips do not contribute a full nutrient profile to the soil, and will require supplementation. This is especially true the more fresh the chips are.

Here is an article that will help you to amend your soil: https://www.gardeningknowhow.com/special/organic/the-importance-of-soil-for-an-organic-garden.htm

Leaf mulch is generally good for your soil, so digging it into the soil in the spring should be fine.

Here's a good article on leaf mulch:

https://www.gardeningknowhow.com/garden-how-to/mulch/mulching-with-leaves.htm

Hay contains seeds, and will likely cause a different set of weed problems. It can still be done, as it makes a decent mulch if you can keep it from getting an infection. It just isn't the best for weed prevention, since it seems to cause as many problems as it solves.

When you use plastic to kill weeds, you do not leave it down as a barrier. This is TERRIBLE for the environment. It is mean to be taken up after it kills the weeds, then you you use the correct fabric weed barrier before laying your mulch.

Here are some articles that will help:

https://www.gardeningknowhow.com/garden-how-to/mulch/best-mulch-for-garden.htm

https://www.gardeningknowhow.com/garden-how-to/mulch/benefits-of-using-mulch.htm

https://www.gardeningknowhow.com/edible/vegetables/vgen/vegetable-weed-control.htm

You can use rocks, bricks or lawn edging; lawn edgings come in many sizes and types available at most garden centers or even big box stores.

I have used different mulches to my gardens for years. It can be done so as to make your gardens more eye-catching. I have used a quilt like pattern at times to create interest. I also use a 3/4 inch decorative gravel mulch in some areas.