To give you a really clear answer, it probably depends on the person. For example, organic grains might be ok for one person, and a definite no-no for someone with a more gluten sensitive genetic makeup. Someone with detox problems (like a compromised glutathione system) might be better off with organic potatoes than conventional apples, while someone coming from a mal-nutrition background might need the extra nutrition density of fruit, and might do poorly on starches since they have very low nutrient density compared to fruit. See what I'm saying? This might take a bit of experimentation on your part to figure out.
You said you have a sensitive stomach, and can only eat certain foods. Based on that, it's probably much more important for you to eat the types of foods that your stomach can handle than whether it's organic or not. Also, as you mentioned, ripeness is an issue. If I have access to ripe conventional bananas and under-ripe organic ones, I'll eat the ripe ones every time.
Also, check out the "dirty dozen" list of fruits/veggies, and try to concentrate on eating organic versions of those.
I really don't know about properly cleaning produce. Not sure if vinegar, bicarb, or store bought fruit wash spays are any good.