Volkswagen Polo ignition/immobiliser key problem

Just bought a 2002 Polo TDI. After about a week or so, the ignition immobilizer light (image of car with key inside it) would intermittently appear and prevent us from starting the car. We would wait a few minutes until the light stop flashing and the car would start normally. Over the next couple of weeks, the light appeared more frequently and would take longer to go away. It reached the point where we couldn't start the car for several hours. Took the Polo to VW. They changed the ignition mechanism for 400 Euros. The car worked fine for about a month and now the light has started flashing again and preventing us from starting the car. Took car to regular mechanic who did a diagnostic and corrected a malfunction that had something to do with a weak signal from the key. They think maybe the problem is related to the fact that the keys do not have a remote function (probably a previous owner lost the original key), which for some reason causes the malfunction. Previous owner says they never had the problem we are experiencing, although sometimes the heater wouldn't turn off or lights would appear for no reason. Actually, sometimes the AC blows hot air instead of cold, which seems to be a common problem judging from the postings I've read.

We are convinced the problem is not going to go away despite the latest software repair. I would very much like to know what could be causing the problem. Any ideas?