For onions, I'm with some of the folks above - I have been presented with them in ways that I find quite literally disgusting, but cooked other ways they are fine.
However, I feel about peppers, all peppers (excepting the "peppercorns" that make the ground black or white pepper on the table) the way you claim to feel about onions. As such, I'd recommend not using them at all, at least in anything that you are eating. If you feel that you need to cook dishes for your family that have them, cook a separate dish for yourself that does not, or cook the onions separately and add them to the dish after removing part of the dish for yourself. Once peppers (very much including green bell peppers) have contaminated a dish, it's useless to me - removing the source of the foul taste does not remove the foul taste throughout the dish.
For your three point question:
A) Nothing else is really a direct substitute, and I suspect that most will be too close if you feel this strongly about onions; ie, I'd be surprised if you like garlic, leeks, or shallots, all close relatives of onions.
B)You can leave them out if you are an adaptable cook. If you are inflexible and following a recipe that depends on the liquid in the onions, you might have issues. If you see that things are too dry and compensate, should be fine, other than...
C) It's for the (highly variable, depending how they are prepared) taste. But if you don't like that, it's not a good reason to use them, is it? If, however, you have an objection to one part of the flavor/texture spectrum that they provide, but not others, preparation/cooking becomes key. But I know that there is no method of preparing or cooking a pepper (other than throwing it in the trash, not in the food) that makes it taste good to me; and if that is your relationship to onions, skipping them is your best bet.