Working with oak means keeping the colors warmer...so if you choose white...they need to be off white or even a creamy ivory white. But that would work. The gray is too cool for the oak.

Hi---When using gray with wood, I find it most appealing if the gray is a saturated hue with a warm undertone. Here are a couple of color palettes for you. BM's Gettysburg Gray paired with French Canvas or Gray Huskie paired with White Dove. Could do light walls and gray cabs or vice-versa. Both examples look great with wood... Hope this is helpful. Good Luck!
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Consider the wood trims a neutral or an accent. You can blend them in with countertop colors, backsplash or accent. But don't let them dictate your style. At the end of the day they are just a trim color and will only be a side note when looking at the finished product.

The attached photos are of a very elegant Newport Beach home that I did a couple of years ago that had wood trim windows throughout.