A correct answer to your question will require consultation with the administrator of your Mathematica license at your academic institution. Wolfram has a number of options licensing options available to academic end-users. I have experience with administering an academic site license and can provide some insights but you should speak with someone at your institution to get the correct answer.
Will I lose access to my mathematica version? Yes, eventually. Student licenses that allow the software to be installed on personal computers typically require annual renewal. The new key will be sent to your university email address. I assume you will lose access to that email address upon leaving the institution.
Will I be forced to upgrade and pay for a different version? No, unless you live in a country that mandates the usage of certain software packages by its citizens.
Is there a periodical check from Wolfram on my student status after say a year? Yes, for license options that include a home-use rider. A typical implementation is that a student creates an account on the Wolfram Portal that is linked to their institution's email address. Then the student version of the software will expire after one year, requiring the student to re-activate through the portal. As mentioned in the first question, if you cannot access the email account provided to you by your institution, you will not be able to access the activation key provided through the Wolfram Portal.
Note added in proof: As an example of licensing customization, this institution provides a decent discount allowing students to migrate their "free" student edition to a professional edition. As a license administrator, I was not able to offer this type of advantage to students at my institution.