I am a graduate student and I had asked for TA/grader position, with a few professors at my institution in fall semester. But at that time there was no position available. Now I want to send another email to them again, asking for a position in spring semester. Should I reply back to the previous email that I had sent, or should I send a new email? How to remind them that I had approached them earlier?
This is a highly personal opinion due to an individual's emailing habits, but I would not attach pre-existing email on something more than a month old. If you talk to the professor about the opportunity in the future, just add relevant information to the current email. No need to add the entire backstory, unless it has important information that cannot be conveyed otherwise.
"Hello, I had previously inquired about such and such back in the Fall and nothing came of it. I was wondering about any potential such and such for the upcoming semester."
In my experience, in the U.S., whether to provide support to a student, through a teaching assistantship, is decided at a department level, and the person to approach is the administrator who supervises the teaching assistants. The department secretary could clarify for you exactly who this would be.
Here's what I have seen: there are a certain number of students who are guaranteed TA work each year. But when the department finds that there are still some uncovered courses, and funds are sufficient, they give support to some unsupported students as well, and these decisions are made on a year by year basis in late August. I can imagine that in some departments this might occur on a semester by semester basis.
When you write your email, proofread it very, very carefully.