Question 1 - I think I understand the part where a person disagrees with the gender assigned to them at birth but when I see a transgender person, they seem to be striving to dress and look like the opposite gender.
You’re sorta conflating gender expression with gender identity here. Gender expression is the outward way one presents to society. Your gender identity is part of how you relate to those around you and society at large. A good example is with cis male drag queens. In drag, they’re presenting with a feminine gender expression, but they maintain their cis male gender identity.
What I mean by this is I rarely see a picture of a person choosing she/her but dressing and having hairstyles more associated with their assigned birth gender.
Further, with trans people, there’s an extra layer of scrutiny from our transphobic society. Trans people are pressured to appear as though they fit a binary gender expression in order to be recognized as their gender identity. (This also extends to those who fall outside the binary, they are expected to display an androgynous gender expression)
However there definitely still are gender non-conforming (GNC) trans people, just as there are GNC cis people, despite those pressures.
Does this mean that although they were born with certain reproductive organs at birth normally associated with a particular gender, they feel that some part inside them(soul, mind, etc) feels they should have been born with the opposite socially constructed gender?
This one is kind of fuzzy, I think every trans person kind of feels their own unique way about it. For example, some trans women keep their genitalia as-is and some trans men are fine with keeping their breasts and these things may not cause them as much dysphoria as it might another trans person. There is also a social aspect to it, in that if one person wants to be perceived as one gender, they may also want their “expected” genitalia, post-puberty body, etc. Note that those who chose not to medically transition are just as trans as those who do.
My second question and this is where I swear I am not aiming to offend. I will try to explain what led me to this thought - When a person chooses to take hormones that their body doesn’t make on it’s own or chooses to have surgery to rebuild sexual organs that they weren’t born with or to add/remove breasts, Is this element of trans life considered a mental illness?
What you’re describing are various steps of medical transition; to answer your question, no, medically transitioning is not a mental illness. There is mental illness in the trans experience however, and that is gender dysphoria.
Gender dysphoria is the word for the mental, psychological, emotional, and sometimes physical pain, anguish, or torment trans people may feel prior to receiving gender affirming care. Gender affirming care is considered to be the cure for gender dysphoria and may come in different forms: it can be medical transition, social transition, a good support system, therapy, or any combination of these things and more.
The only reason I ask this is I remember watching a documentary where people lived a life in which they felt, for example, that one of their arms didn’t belong to them and they pursued surgery to have a working limb removed. During the documentary, some of the people during therapy and medication were able to change their mindset to the point that they could live with the offending limb but there were some people that were traveling to other countries to have it removed (the doc was based in the US and they couldn’t find a doctor to perform the surgery). The only reason I ask is because of that, My mind goes to body parts that the person doesn’t feel belongs but that they were born with and not something socially attached to them.
A better comparison would be if you had a third arm, or if you were short one arm due to a genetic anomaly or injury. You may feel social pressures that you may not fit in or maybe you feel body dysmorphia about your body being different from other people you identify with. You likely would want to seek care to more closely align your body to one that is socially expected.
Never feel bad about asking questions! You’re making an attempt to learn and understand, and that’s way more than some people (allies included) ever do.