The following is touched on in some answers and comments, but I feel is worthy of a separate answer.
The primary purpose of assignments and exercises like this is not (simply) to get you to produce the correct answer but to demonstrate to your instructor that you both understand the concepts involved and can convey your understanding of those concepts.
Starting with a false statement you can "prove" anything. Starting with a potentially false statement (or, at least, one whose truth is unknown) will lead to an erroneous proof unless you are able to say (and do say) that each step (ultimately leading to a "known true" statement) is an example of if-and-only-if or implies-and-is-implied-by ($\iff$).
Thus, in your answer, even though – as you say – "Dividing by 2, adding and factoring are pretty obvious right?" because you have not explicitly stated that each step is reversible, your instructor does not know whether:
You are aware of the dangers of a "proof from fallacy" but have not bothered to make this clear, or:
You are blissfully unaware of those dangers, and – while you got lucky this time – you might use $-5=5 \Rightarrow 25=25$ next time and not notice the proof is invalid.
In a similar vein, when I was at school we were always told to show our full working, because only then could the teachers determine whether we understood what were doing or not. There were several outcomes:
Incorrect answer; no working: Zero marks.
Correct answer; no working: Limited number of marks (probably no more than half the total).
Incorrect answer; workings shown: If the workings demonstrated the correct method, but included a simple mistake, you were likely to get reasonable marks (possibly three-quarters).
Correct answer; workings shown: Assuming the correct method was used, full marks.
To answer your specific questions:
- Should I have received marks for this question?
(original question; as originally answered)
In this case, I'd be tempted to agree with your instructor. Having chosen a "non-standard" layout, and without explicitly indicating you are aware of the dangers of a "proof from fallacy", while a little harsh, I don't think zero is overly harsh. The fact that it has stuck with you sufficiently to come here and ask about it is evidence that you won't make the same mistake again which is really the main goal of assignments like this.
- Is my proof equally valid?
(after question edited by other than OP, although I believe the original question is more relevant)
Other answers directly tackle the "correctness" of the OP's proof and I don't really have anything to add (other than a personal view that the assumed if-and-only-if nature of each step should have been made more explicit).
The point of my answer is that this question is not what someone in the OP's position should be asking (and, to be fair, the OP didn't actually ask this question). What is important is not whether the proof is correct, but whether the student has demonstrated their understanding of the topic.
So: the proof may be correct, but it's still a bad answer to the exercise.
- Do real mathematicians all write one way or the other when writing in a paper?
I am not familiar enough with the writing of academic mathematical papers to be certain, but I would expect (hope?) that were someone to present a proof in the form you did, they would be explicit in their use of $\iff$ between stages (or, more probably, would have used the "conventional" layout).
(An addendum after responding to the edited question.)
The premise of my answer is that tests/exercises like this are predominately about a student demonstrating that they understand the topic at hand. When I was at school in the UK, studying for what then were O Levels, this was drummed into us constantly: getting the right answer is less important than showing how you got it.
You may possibly have a legitimate cause for complaint if your instructors have not sufficiently emphasised this point in the past.
If you feel that this is the case, I'd recommend a conciliatory (rather than confrontational) approach. Explain to your instructor that you now understand why they felt unable to give you marks for this question – i.e. because your approach leaves room for not understanding the problems of proof by fallacy — but that you feel this need to demonstrate such understanding hadn't been sufficiently "driven home" in the past, and could they (the instructor) focus more on this aspect in the future.