Mushroom Vent Caps can be attached to any vertical sanitary vent pipe, in order to keep debris from falling into the pipe. This doesn’t apply to septic tank vents which rise vertically out of the ground, and then turn down (looking like a candy cane).
No. This product is not intended for use in any plumbing system where plumbing codes are enforced. 
No. As the cable and cutter pass through the valve, the flapper would be damaged. Once the cable passes by the flapper, it would not be able to return because the flapper would close down on it. This would cause permanent damage to the flapper. Cleanout Tees should be installed on either side of the Backwater device.
You will have to pay attention to the drains for signs that the Backwater valve is closed. If the valve is closed the water will not be able to pass through it and will back up through the lowest drain in the home (floor drain or shower base in the lowest level of the home). The drains will also gurgle as water is being let out of the fixture.
Yes. After every heavy rain storm it would be a good practice to open the access cover and make sure no debris was lodged in between the flapper and body, which would not allow the flapper to close. If there are not any storms in a 3 month period of time, it is also recommended that you inspect the flapper to make sure that no debris from normal use is lodged or stuck between the flapper and valve body. This will ensure the valve can seat properly if a storm passes through. 
No. The flapper would remain in an open position without any backflow protection.
You should contact the barrel manufacturer for their recommendations.
You can disconnect the hose and insert the red plug. 
If you disconnect the hose and insert the red stopper plug into the outlet of the diverter, the minimal amount of water that freezes inside the diverter body will not damage the diverter. The diverter is designed to redirect a small amount of water into the outlet as it spirals through the downspout. Once it is capped off a very small amount of water will be trapped and able to freeze. This small amount of water will have plenty of room to expand as it freezes without causing expansion damage to the diverter. 
100% silicone caulk is recommend. There are many conditions where plumbers putty could cause product failure and finished product staining, please refer to category #4 FAQs for these applications.