There are only a few rare situations where it may be safe to make yourself throw up, such as in cases of mild poisoning from a non-corrosive substance. Even then, you should never induce vomiting ...
For the phrase "putting yourself together," I can see that it doesn't mean controlling your emotions to deal with a situation. So, I did some research and found that "to put oneself together" can mean making one look nice, maybe by dressing up or using makeup, which fits the context of the sentence above.
Yourself stresses that YOU (personally), as opposed to others, can be the agent of that choice. By yourself / on your own stresses more that you make this choice alone, without help from others. So although the meanings may slightly overlap, the terms are not interchangeable. Your sentence is about receiving a given topic as opposed to the possibility to choose that topic personally (the idea ...
To spew is to forcefully expel something, the way a volcano spews hot lava when it erupts, or the way you might spew soda from your mouth if your friend makes you laugh right after you take a big gulp.
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Using "yourself" and "ourselves" in these contexts is incorrect. "Yourself," "ourselves," and "myself" are reflexive pronouns, correctly used when the subject/actor of the sentence and the object/recipient are the same person or group. "I see myself" is correct because I am doing the seeing and am seeing myself. In your latter example, the subject is the implicit "you" and the object is ...