Some knowledge is obvious when one does not need to think on it, so, when it raises unconsciously (independently of the underlying mechanism that makes it raise). (see also conifold's link).
If I tell you the sun will raise tomorrow, you take the existence of the sun as a true fact, and don't even think of it. The existence of the sun is obvious for you.
Remark that we're talking about knowledge. Knowledge is subjective. Two individuals might share some knowledge, and have more, which remains subjective.
Remark also that knowledge is basically a set of relationships. For example, "an apple is a fruit" is a relationship between the concepts apples and fruits. When you are learning, you build such relationships. Later, you don't rebuild them every time. You just use a pre-established relationship.
A) How do we know for sure that somethings are obvious while others are not? Does truism partly answer this question (and if yes, then why)?
We don't. You can almost assume that all the members sharing a specific culture have the same knowledge (for example, every member of a group of computer gamers know what a computer is), but that's not for sure.
B) Does the obviousness of a statement (or an answer) depend on whether the context or the premise being used is either deductive logic or inductive logic?
Either knowledge is obvious (one does not need to think on it, because a pre-established relationship between concepts already exists), or either one does think on it (in order to create relationships), in which case logic would be applied. The sentence "you can buy apples in the market" is obvious because you already know that, you don't think on it. You didn't applied inductive logic, or deductive thinking. As said, you just apply pre-established relationships. If you think, you apply possibly inductive or deductive logic, but that's not anymore obvious.