Looks like you will need to substitute vegetable oil for the eggs. There are egg free recipes available.
Yep, vegetable oil is always the best to use when you are baking.
Vegetable Oil Fuel Conversion
Bananas are a good egg substitute for sweet dishes so I'd try that first :)
You have to use some kind of oil. Olive oil is probably the best, although any kind of vegetable or fruit or nut oil will do. The oil combines with the yolk of the egg to form an emulsification. Without the oil, you can't get mayonnaise.
Olive oil is a vegetable oil. You can often substitute other vegetable oils, but only experience, or a knowledgeable person can tell you it that will work for a particular recipe.
a boiled egg
yes.
yes
Yes, but the crackers will have a mild olive flavor.
Yes, but be aware that olive oil has flavors that most vegetable oils don't have. You are likely to have a touch of olive flavor in your pancakes!
In baking, you can substitute apple sauce for most of the oil. For sauteing, you can use butter, but it may burn. For frying you have to use oil