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은/는 is the topic particle. 이/가 is the subject particle. I think adding 가 to "I" would make "I" the thing doing the action in the sentence. 가 identifies who, what, which thing...I believe everything else you said is right. "WHO is going?" "제가요"/"It's ME"/"I am"
Object marker is 을/를. Object is something that is being acted upon.
저 is normal form of I (formal). 제 is 저 +ㅣ so it becomes 제 and means I as subject.
Remember topic vs subject rule? 은/는 for topic and 이/가 for subject.
So to write I as topic, use 저는. To write I as subject, use 제가.
Another example is 나, also means I but informal. 나는 = I as topic. 내가 = I as subject.
In general, both of them are acceptable in most of cases, in grammar. However, depending on the content you want to say, it can be variable. I will give examples based on my experiences: 1. 저는, when I translate, it mean "about me" or "myself". Then, I usually use in description anything about me rather than an action in which I am actor. 2. 제가, it is more prefer than 저는, if you don't know exactly which is correct. I use it in an action state, or say somethings of mine.
I'd much rather be exposed to the 요 form right away. Perfect level of formality for day to day living plus you wont be looked at weirdly by people you meet. I said 만나서 반갑습니다 to one of my Korean friend's friend and he suggested I use the 요 form because it sounded like we were at work and being over the top.