Let me echo Benjamin's comment that any proactive step that you take should be done with the instructor's permission.
At a practical level, I think there are ways to address issues (a) and (b). For (a), make a rubric (either in advance or a running one as you go) which lays out the criteria for awarding points. This allows you to be consistent with how you distribute points and, since the assignment of points itself is subjective in nature, a reasonable rubric enforced as consistently as possible usually results in good grading. Your instructor can give you a sense of whether your overall grading standards are too harsh or too lenient. For (b), if you have followed (a) and have the support of your instructor, I think it would be a reasonable policy not to consider changes to homework grades unless a substantial error was made (e.g. you misread a student's response and marked a correct method as being incorrect).
But regardless of grading issues, we would certainly like calculus students to produce better responses than the one you used as an example. In my view, one needs to take a perspective similar to triage in medicine. Often, students have many deficiencies relative to the level of performance we would like to see from them, so prioritizing the most important issues to address is essential to good instruction. Since examples are often easier to discuss than general principles, I'll stick with your specific example.
I see one glaring issue that is very concerning, and that is the use of the step $\frac{\infty}{\infty \cdot \infty} = \frac{1}{\infty}$. Not only is it an invalid thought process, it is likely to produce incorrect conclusions if the student attempts to extend it to other contexts. Would $\lim\limits_{x \rightarrow \infty} \frac{(x - 1)(x + 4)}{x^3}$ equal $\frac{\infty \cdot \infty}{\infty}$ or $\frac{\infty \cdot \infty}{\infty^3}$? What would happen if the denominator is $\sqrt[4]{x}^3$ instead of $x^3$? It's not clear that the student would use the right "power" of $\infty$ to be able to compute the limit. (If they are able to, that's a positive sign -- at least they have some sense of what is going on, even if their articulation of the reasoning is very flawed.) In comparison, if a student writes $\lim\limits_{x \rightarrow \infty} \frac{1}{x} = \frac{1}{\infty} = 0$, I'm not thrilled with the inappropriate use of $\infty$ (according to the standard definitions), but I consider it mostly harmless. There is reasonable sense that can be made to $\frac{1}{\infty} = 0$ that is much more palatable than nonsense such as $\frac{\infty}{\infty \cdot \infty} = 0$.
The other issue that I would want to address is that "H.A. at 0" is not very precise and I would prefer a student write that "the line $y = 0$ is a horizontal asymptote" ("H.A." in place of "horizontal asymptote" is fine), but some instructors may prefer the less formal means of expression. After all, someone could be pedantic with my approach and say that $y = 0$ is not actually a line, so we should write "the line which is the locus of all points where $y = 0$" or some such and now we've simply confused the students entirely. In practice, I would accept an answer such as "H.A. at 0" from students, but expect myself and my TAs to be more precise and refer to the line $y = 0$ as the horizontal asymptote.
Once you've identified the issues you want to address, here are some concrete ways you can obtain improvements:
Be sure you are very careful in your own explanations of solutions. I've had TAs who wanted to mark very harshly for sloppy writing on homework ... and then I observed them engage in the same sloppiness on the board during their recitation periods. When I brought this issue up, they argued, "Well of course I have to be sloppy in class because I am under time pressure, but students doing homework have the time to write better." The flaws in this reasoning are many, but in brief (1) it is generally better to write explanations clearly on the board and cover less problems than to sloppily solve many problems; (2) students cannot possibly learn to write well if they do not see a consistent high standard [and certainly not if they see sloppiness is acceptable for an instructor or TA]; (3) students do not have the time or do not choose to use their time to write polished solutions to every calculus problem they are assigned -- and it's not realistic to expect that [so if good writing has not become a learned habit, don't expect it to happen].
So you start by providing a good example yourself. Then you perform triage and identify the worst offenses that you will mark off on homework. The cancellation of factors of $\infty$ in your example would certainly qualify. You should take time to address common mistakes. I recommend a brief amount of time addressing such issues in class (a list of things not to do and 1-2 sentence explanations of why for each item on the list), with an offer to explain more thoroughly the rationale in office hours. Students will respond to the incentive of losing points, and they are generally accepting if you can show that there is a reasonable rationale behind it (even if they don't fully understand or accept that rationale). Just don't appeal to authority or superiority. If you cannot give an explanation that a reasonably strong student can understand, you are probably setting an unrealistic expectation.
As a TA, it should be added, that you should generally rely on your instructor's opinion to perform your triage assessment. Certainly it is inappropriate to simply decide your own priorities without any consultation of the instructor. The instructor often cannot micromanage every single prioritization decision you make, but a good conversation with your instructor can leave you with a good understanding of how to prioritize (and a good instructor will understand that at the level of small details, you two may not come to the exact same conclusions and that is okay as long as you are in harmony overall).
Ideally, by the time exams come around, students have had good instruction to see what an appropriate solution looks like for the types of problems they encounter, and the homework has identified the most egregious errors in writing up solutions. Then the markings for exams should generally follow the same standards. (In practice, I prefer to be have a somewhat more harsh grading standard for homework than exams. Harsh feedback on homework is more likely to get their attention and lead to improvement before the next exam. And that improvement will hopefully lead to increased retention of conceptual understanding for use later in the course and beyond.)
