Clearing 2 week old transactions from mempool
What stops Blockchain.info to rebroadcast my transaction to the
  network again?


Nothing. Or may be laziness.


  It would be exactly the same as if I just broadcast my transaction for
  the first time, right?


Right.


  If Blockchain.info can do it, is this 2-week rule true in the first
  place?


Everyone can do it. And someones definitely do it. Your node will get it back after 2 weeks.


  If it is true, how it is implemented on the network level?


It is not consensus rule. If you want - you have a right to rebroadcast a transaction if it is still valid [and not confirmed].


  Is it some kind of agreement that we do not rebroadcast transactions
  that are this days old?


No.