That is possible, but it depends upon how bad you are at essay writing. You should probably ask your tutor to evaluate your essay writing and give an opinion of it. Hopefully you have a good tutor who is capable of doing that.
In most cases, no, but it will depend on the instructor. It shouldn't though.
Classes in writing are are typically required to earn a degree in science and engineering. Those classes determine whether or not a student has sufficient writing skills for science and engineering careers.
Math is good for science and engineering of any type.
No. If you're bad at essay writing, you might as well just get used to saying "You want fries with that?".
Clear writing is a valuable ability regardless of what vocation you choose, but the lack of this ability will be more of a handicap in the liberal arts (English, literature, history, etc.), so in choosing the lessor of two evils engineering would be a better choice than liberal arts. Science requires writing many scientific papers, so demands writing skills more that engineering does.
Yes
Writing an essay about why science is important will teaching you why it is important. You will learn how to actually use science in your life.
In Technical Writing, we eliminate technical jargons and write in such a way that it is understandable by laymen. We need to make sure the meaning of the original technical word doesnt change. Non-technical Writing doesnt include or relate to technical stuff. This covers a vast range like - essay writing, blogging etc
Each of the skills mentioned in the question is important in different ways. While each or all might be important for an engineer or technical report writing, but that can only be determined by the requirements of the report. Now, essay writing in academics (school) is necessary for practice. What is critical for all writing is the ability to get your message as the writer across. If essay practice helps with this, then great. If programing skill or chemistry knowledge do as well, great. Better to have tools you never need, than need and no tools.
Being able to present your findings is important, but understanding the material is more important. The trick to essay writing is to be as concise as possible while still giving a deep enough answer for the full award. While I would say that essay writing skills are important, it should not be the end of your carreer.
The context of an essay, specifically one that is a writing in your chosen field, would vary depending on the subject.
Any field will eventually pose a challenge for someone who is weak in essay writing. The best solution for the problem is to improve your writing skills, and the best way to improve your writing skills is to A) take more writing courses than simply those required for graduation, and B) write as much as possible - essays, articles, papers, etc. Never simply resign yourself to being bad at essay writing. If you think or know that you are, do what is necessary to remedy it.