You know, it's rather hard to say anything helpful when I don't know exactly what's running on your computer.
I still don't see the connection between the DOS program and Internet Explorer (which is certainly not a DOS program). Are you saying that you use the DOS program to enable ("start") the while Internet access on your computer? (Rather weird, I must say... what kind of program could it be?)
What are your settings of the Standard Shield provider?
Second, what has a .lnk extension? The DOS program itself (another very strange thing) or the .lnk file just points to the DOS program (in that case, would it make any difference if you start just the target program)?
In any case, I don't see any way how avast! could cause anything like that. The Standard Shield provider will check the file when the program is about to be started - and if it's clean, it's simply allowed to go on - and no other (delayed) actions are taken. So, I don't know how it could stop the program 30 minutes later (well, unless the program is trying to do something bad, like extracting a virus somewhere?)
You can try to Stop or Pause the particular resident protection providers of avast! to see if it helps. If you want to prevent scanning your file by the Standard Shield, you can just put your program to the list of exclusions (click the avast! blue tray icon, click "Details..." if not already, doubleclick the Standard Shield icon and move to the "Advanced" page - you can enter the location of your program there) - but I don't think it will help...