Most importantly, what would be good woods for a table mat?
This doesn't necessarily have to be about the choice of wood because your finish alone could cover you on this front.
Heat and moisture-resistance are generally taken care of by finish, because even the most stable of woods may not respond well to being heated and/or dampened on one side only repeatedly especially when the material is thin.
I don't want them to distort with age,
I can virtually guarantee they will if you make them from solid wood.
Asking for 9mm material in a board of approximately A4 size is asking a lot of wood, even a stable, strong hardwood carefully chosen to be QS might struggle.
If you're dead set on using solid wood for this and on this thickness I would strongly recommend these are stacked vertically if wiped with a damp cloth even if you fully varnish them, which in practice means at least 3-4 full-strength coats or the equivalent wiped on (7-10). Note this would be on both sides and all edges, not just on the upper face.
A lamination would be a much better idea here, either plywood or a lamination you make yourself. Inherently more stable, much stronger in both directions and it will also make the inlet decorative bands easier to do......
Grain orientation in the decorative strips
I'm not sure if you'd thought about grain orientation in the inset bands in relation to the main body of the mat but at a guess you were intending to do them cross-grain. And if you do you will have issues, and they'll probably show up much sooner than you like.
The way to get around this is to make cross-grain banding, but fair warning, making it is tedious and potentially difficult.
Easier to have the grain of the mat run vertical, same as the strips.
Should I choose a contrasting wood for the dark strips, or just stain some strips?
Staining strips like this and not having the stain bleed into the wood around them is difficult or impossible1. So I think this means inlays of contrasting wood picked itself :-)
I'd like to put a food-safe finish (not sure what I should use though)
Purely in terms of the food-safe aspect anything is suitable really. This can mean literally anything you can think of, from vegetable oils (cooking oils2) all the way through to epoxy and everything in between, including BLO which on paper is toxic but in practice is not.
But this shouldn't be your primarily concern as I touch on above, protection of the wood should be unless use a lamination in which case it opens up a wider range of acceptable options (including no finish at all).
If you go with solid wood I would highly recommend going with oil-based polyurethane. If that's too yellow for you I would pick a quality waterbased poly, but do be careful to pick a good one as they vary a lot in how well they protect wood.
1 Depends to a degree on species and stain type and whether you'd be OK with heavily knifed dividing lines.
2 No they won't go rancid that you'd notice. People were oiling salad bowls and utensils, even chopping boards, with vegetable oils for generations before the current craze for mineral oil.