(1) I agree with you. Offer, like give, takes two objects, a direct object, and an indirect object.
It's possible your instructor was making a distinction between double object constructions ("ditransitives") like "offered him a scholarship" vs a transitive+to construction "offered a scholarship to him". (There's also arguably some difference in whether the scholarship was actually transferred/given in this case, but I don't believe telicity [whether an action is completed] would determine its valency [how many arguments it takes, e.g. 2 for ditransitives].)
Your instructor either made a mistake while grading or has a special reason for this in a particular use of the terminology. Note that terminology can be used very differently depending on your book, your instructor, and even specific purposes at any given time. It's a real problem, because it gets in the way of really understanding what's going on when the focus is on the terminology, but on the other hand we need some way to talk about things.
(2) I don't know what a "complex-transitive verb" would be. Sure, it might be that. Or it might be something else. It depends on the definition. This is some sort of secondary predication (in the most usual terminology, I think), where "[be a] flop" is a secondary predicate for "movie". I want to say depictive, but I must not be thinking of the right specific subcategory because that's describing the object, not the subject (a common example of a depictive is "I [did something] naked").
(3) Your answer sounds reasonable. But again it depends on terminology. I'm not sure I'd strictly consider that to be a relative clause because "rumour" has no role in the embedded clause. A relative clause is usually one where the relativized noun has a role like subject or object in the clause ("That's the man who bought the house" -- subject; "That's the house he bought" -- object.)
I think a common term for this might be "object clause", although it's a little weird that it has "the rumour that" rather than just "that". It's similar to "The fact that", which is more common and might be discussed in your textbook if you want to find the term.
I'd ask your instructor about these-- they're not bad answers, even if they aren't the preferred answers. They may also be able to suggest a better way to study (e.g., where the exact definitions are, since there are many terms and definitions you might come across elsewhere).