Starting from September 1, local new students will begin their first studying emesterm of thise school year, and will end by January of next year.

Depending on how many 学期 there are during the year, there are different words such as semester (two), trimester (three), and quarter (four). In the original sentence, you imply that "new" students 今年都没有上学, although I think you just mean that it is their first semester at a new school. The edit makes the sentence a more general statement about when local students will start school. Maybe I misinterpreted, please correct me if I'm wrong!

As a new student in seniofirst-year high school student, my daughter was asked to attend a mandatory military training program for a week.

I'm American so I don't know much about the British education system (except that there is no "senior high school"), but here there is elementary school, then middle school, then high school, which can be generally correlated with 小学/中学/高中学, although high school lasts four years (高三=fourth year of high school).

Each of themstudent will have to pay 510 Yyuan (around $72), includingwhich covers three meals a day and clothes for training.

Currency names such as yuan or dollar are not capitalized.

Additionally, anyone who is not able to join the training for physicalparticipate in the program because of health issues also needs to pay the fees, or else they won't allowed to graduate.

You can use additionally or also, but not both. physical issues -> health issues sounds more natural.

By the way, please pay another 300 Yyuan as an expenditure reserve for class daily usagefor daily class expenses.

Hhh, they are really asking for a lot!

Starting (from) September 1, local new students here will begin their first studying term of this school year, andwhich will end by January (next year).

“Local” is unnatural in this context.

As a new student in senior high school student, my daughter was asked to attend a mandatory military training for a week.

Each of them will have to pay 510 Yyuan (around $72), including three meals a day and [clothes for training | training uniform].

Additionally, anyone who is not able to join the training fordue to physical issues also needs to pay the fees, or (and join a follow-up training), or they won't be allowed to graduate.

By the way, pleaseeveryone also has to pay another 300 Yyuan as an expenditure reserve for [class daily usage].

“For class daily usage” is wrong, but I’m not sure how to fix it because I don’t know what you mean.