I'm not sure what you mean by saying that "感情 can't be the subject of a causative verb". Since 感情 is marked by を here, it clearly isn't the subject of 昂らせる, it's the object. (The subject would be the implied 私.)
Perhaps you meant to say that it shouldn't be the object of 昂らせる because this is equivalent to being the subject of the non-causative 昂る, but in this case you are mistaken - 感情 is a perfectly natural subject for 昂る. The verb 昂る refers to "something" becoming intensified to an extreme level, and as such 感情 is an entirely natural and common subject for this word. Indeed, the dictionary entry you linked for 昂る lists 気持ちが昂る as one of its main examples, and 感情 is almost identical in meaning to 気持ち.
So if 感情が昂る literally means "my emotions become intensified", the equivalent causative structure 感情を昂らせる in this sentence literally means "I cause/allow my emotions to become intensified". This isn't a particularly natural expression in English (since such emotional reactions aren't really voluntary), but it's common to use causative constructions like this in Japanese as a way of making the person involved the subject of the action.
Moving on to your question 2), I don't think the phrase 自分を昂らせる would be as natural here, because 昂る usually takes an emotion or aspect as its subject, not a person. But it's not an impossible construction, and I believe 自分を昂らせる would indeed probably be understood as having a roughly equivalent meaning to 自分の感情を昂らせる.
As for your question 3), if you removed the 感情を, the sentence would be less clear and arguably ungrammatical. Like all causative constructions, 昂らせる needs to have an object to make any sense, so if there is no object in the sentence, an implied object would have to be assumed. The verb 昂る is so closely linked with the concept of emotion that it might be possible to assume an implied 感情を even if it's not explicitly stated (or perhaps you could assume the previously-mentioned 怒り as the object), but either way it seems clearer to include the object explicitly.