so my mother moved my siblings and I
-- Correct me if I'm wrong, but I think the word should be "me". I think, in this type of sentence, where there is a conjunction, the personal pronoun takes the accusative form.
I hadn't made a connection and settled for the fact -- A comma seems necessary, just before the conjunction.
I had no vehicle, but managed to get around the city.
This sentence is not necessary. This is better explained in the second paragraph.
they warned me that it would be challenging and expressed that they were confident I would change my major. My confidence was lowered further
Here the two ideas seem to be in contrast. Did they encourage you? If they did, then why would your confidence fall?
I see that the placement exam lowered your confidence, but why was it low in the first place when your parents had encouraged you?
Thus far, my calculus and physics courses have been the highlight of my college experience. I am anxious to continue my studies and take classes covering subjects such as thermodynamics, mechatronics, and materials engineering. I recall one of my professors explaining that given the right tools, one can solve any problem. I am excited to acquire all the techniques and knowledge necessary to apply to real-world issues.
I thought this paragraph was totally irrelevant to this essay.
As an altruist, I can't help but pursue a career in which I will apply my knowledge to improve the quality of life of others.
This is a weak sentence. You shouldn't say you're an altruist. Make the readers conclude themselves. The rest of the sentence is stuff that anyone else can say.
I have to say you write well. However, this essay doesn't address the prompt completely. You didn't say a lot about biomechanical engineering. You should also be specific about your career goals. Biomechanical engineering is a large field; can you say something specific you want to do in this field? If you have done the research, you'll know what to write.
Good luck!! We have the same field of interest :)