I am looking through au's website, and I see that I can only c-mail other au phones, but can I use e-mail to contact phones from other providers?

If I understand correctly, to have e-mail I must be subscribed to EZweb. The charge on au website's is a monthly fee of 315 yen. Other than that you have to pay per packet of information sent. I'm trying to understand, if a packet costs 0.27 yen, how much will an average mail message cost me? Does it matter which service provider's phone I am sending the e-mail to?

Also, about student discount, will it cut my costs even if i'm on the simple plan? meaning the 1,050 charge will become 525.

Email is more common. C-mail (essentially an SMS) has a length limit and is restricted to the AU network, while Email has no character limit and can email other network's cellphones and computer addresses. However, if you have friends or family with AU phones then you may prefer to c-mail them as it is free (depending on your plan).

I am looking through au's website, and I see that I can only c-mail other au phones, but can I use e-mail to contact phones from other providers?

yes

If I understand correctly, to have e-mail I must be subscribed to EZweb. The charge on au website's is a monthly fee of 315 yen. Other than that you have to pay per packet of information sent. I'm trying to understand, if a packet costs 0.27 yen, how much will an average mail message cost me?

Just a few yen. A 100 character email would cost around 1.7 yen. Its actually cheaper than c-mail if you don't have a plan that includes free c-mailing.

Also, about student discount, will it cut my costs even if i'm on the simple plan? meaning the 1,050 charge will become 525.

I'm not sure about the student discount restrictions but it very well may discount it to 525 yen/mo. Be aware to qualify for the simple plan means you have to buy your phone outright. Phones generally start around 30,000 yen.