The jacket is dark brown with barely visible light greenish-blue checks.
Would a plain-white shirt go with it? What about deep oxford blue? What about a white shirt with thin widely strapped black straps? What about a light blue stripe shirt? Stripes though, probably don't go with checks, do they?
Would a white plain (think mad men, Don Draper) pocket square go with it? If not - what would?
Since it's a sports jacket, I think no tie should be worn with it, and the top shirt button unbuttoned.
I'd be wearing black shoes (no patterns) and either dark blue jeans or one of these trousers (which ones would go with it, and which are more versatile, by the way?):
Appreciate any advice!
I'd probably try to emphasize the greenish-blue in the jacket and choose a very pale green or blue button-down shirt with no stripes. It sounds like all your surrounding components are neutral, so most colors would work as long as they're not too strong.
Look to nature when combining colors. God did a great job with color palettes.
Brown reminds me of bark, like a tree. So, think colors you would find on a tree...green is obvious, but also blue, pink, yellow, lavender, etc.
Though you identified it as being subtle, checkers can be thought of as a strong pattern. I would stick with neutrals, which for you means solids, when coordinating. Extremely subtle patterns, ones which when viewed at arms length while squinting appear solid, might also work, but be careful.