Attaching reprints of published but copyrighted papers with job application
Like almost all such questions, the ultimate answer is "look at the copyright agreement you signed."

However, in almost all cases, it is fine to distribute copies of your papers privately to individual people, so I can't imagine there would be a problem with you sending your papers along with your application.  In the past, publishers would give authors a certain number of paper "offprints" for exactly this purpose.  Nowadays, you more often get an official PDF (with the journal's logo) that you can send to your friends and colleagues, and that would be just the thing to send with your application as it also proves that the paper was in fact published.

Posting on your web site is likely to be specifically addressed by the copyright agreement.  Some publishers allow it, some do not, some allow the posting of only certain versions (e.g. before revisions based on peer review).  So read what you signed.  (And next time you publish a paper, read the agreement before you sign it!)