A splitter will not cause a problem with your computer shut-down process.
Regarding the issue of not being able to access the internet: Without being there I can't say for sure, but I suspect one of two problems.
1. The capacitor effect. Where coaxial cable with a brass connector is connected to a splitter made of aluminum, if ANY water, even from the natural humidity in the air, gets in the connection the dielectric effect of dissimilar metals will cause a type of corrosion. That corrosion will create a capacitor which interferes with the signal. When you disconnect the cable from the splitter the capacitor discharges, so the signal issues go away when you reconnect. The capacitor effect will be more pronounced if there is current on the shield (which I will address in #2). To diagnose simply disconnect the cable from the splitter then immediately reconnect. If the problems go away for a while, you're experiencing that capacitor effect. Fixing it can be trickier. You would probably want to call the cable company and arrange for a service call.
2. AC voltage on the shield. If you have a bad power ground, and if some device like a VCR or TV also has a grounding problem, then the some or all of the stray current will travel on the coaxial cable shield, which is grounded at the pole or pedestal at the street. That can play havoc with low-quality splitters. You say that the cost of replacing the splitters is not much, but a good splitter will cost around $10 retail, so i suspect you're not using high-quality splitters. Of course, the real problem isn't the splitter, it's the current on the shield, which should not be there. This is easy to diagnose with a volt meter and basic electronics knowledge. If you don't have that knowledge, you will need help. Again, a case where you should arrange for a service call. A cable technician can determine if you indeed have a bad power ground, and identify the bad equipment that is feeding that voltage onto the coaxial cable. He or she cannot fix the power ground, however. That is a job for a qualified electrician. This can be a potentially dangerous situation, and needs to be repaired.
Bottom line- You need to call the cable company. The real issue in not the splitters failing, it's electricity making the splitter fail.