First I had to add Logging to Event Log for the events of level Warning and higher. This did not provide much help but at least I started seeing the Errors in the Windows' Application Event Log.
Searching over the web for similar experiences gave me a few ideas to try out and the one with stopping and restarting all the services on server worked out.
Going to Central Administration -> Topology and Services - Services on Server and shutting down the services seemed to do the work.
However, restarting Sharepoint removed the header mappings for the Sharepoint site. Since Sharepoint took port 80, and there was another server there, I had to add header mapping for the two sites so that the previous web site would still be accessible by its DNS name. Also, restarting the Sharepoint stopped the Default Web Site again.
So, fixing the headers on the Sharepoint site and starting the Default web site manually fixed everything.