Welcome to the forum.
Since the outage seems very consistent and not random when it occurs it seems like a setting could be wrong causing this weekly outage, does this sound familiar?
No, I don't recall any occurrences where the TL-250 caused an Internet connection to drop out. I'm not saying that it's impossible; however, I would explore other possibilities first.
Yes, DLS 2002 provides remote access via theTL-250 and can be used to arm/disarm the system. That said, keep in mind that while it "can" be done, that is not the intended application, it would cumbersome and could easily lead to arm/disarm errors. I wouldn't recommend it for that functionality.
IF you have a landline, the Escort module was specifically designed to provide that functionality and would be far more user friendly.