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The dyes used in commercial mulch is considered pet and kid safe.
Dogs that eat mulch is another issue and perhaps a call to your vet would ease your concerns.

Here is a link with more information.

https://www.gardeningknowhow.com/garden-how-to/mulch/keeping-mulch-safe-for-pets.htm

Here are two great articles that will help you.

https://www.gardeningknowhow.com/garden-how-to/mulch/common-mulch-fungus.htm
https://www.gardeningknowhow.com/garden-how-to/mulch/garden-mulch-problems.htm

I have listed a few links for you to address some of your concerns.
Some years ants can be more of an issue then others.
I do like the benefits of Cedar Mulch, as it can deter some insects.

https://www.gardeningknowhow.com/garden-how-to/mulch/best-mulch-for-garden.htm
https://www.gardeningknowhow.com/plant-problems/pests/insects/how-to-control-ants-in-lawns.htm
https://www.gardeningknowhow.com/garden-how-to/mulch/bark-mulch-in-gardens.htm

Using hay is fine BUT you need to know what's in it, as many hay bales are filled with weed seeds that can pop up in the garden. Of course, if you don't mind plucking these as soon as you see them (and if it's thick enough you may not get any), then by all means take advantage of your freebie!

Shredded bark, rather than bark chips, usually does okay in windy situations. There are actually stones/pebbles that can be obtained in various colors, depending on what you're looking for and I'm sure a brown color would be available too. You might even look into a low-growing ground cover and stepping stones if you want a more natural feel to the area.

I think as long as you keep it in a dry, sheltered location, it should be fine until next season. The main thing with bagged mulch is to keep it dry so that it doesn't sour. Most of the time bark mulch have been dried but if it's left in a humid or wet area, condensation could build up and that's when you would have a problem. So as long as you can keep it dry and cool, it should be fine.