Regarding the intel fortran licence, I read that after the licence expired you lose support and cannot get new updates but you can still keep using the product. Does it mean that I can install the product on a new computer? (Suppose I downloaded the compiler when my licence was still active and now I want to install it on my new laptop)
Yes you can. In the past, the End User License Agreement said something about a "reasonable" number of installations, and I know that if you tried to activate more than 3 (I think), you had to ask Intel Support for more. But now I see it says "You may run as many Instances as needed for Your exclusive use"
Using the license file is probably the most straightforward way of doing the reinstall. It doesn't matter when you do this, as long as you don't try to install a version newer than the end of the license support.
A bit of terminology - Intel licenses (at present) don't expire (except for evaluation and student), but the support term expires and that's what you referred to.