I'll assume that you have an oil boiler. These can be set up in many different ways and depends on what kind of boiler you have. Can you provide more information about your boiler? Take some pics of it and the associated pipes, if you can, and post them.

This same 10 seconds hot then cold again happens at how many faucets, all of them? Do you have mixing faucets or separate valves at all of your sinks/showers/tubs?

Sounds like you have a tankless coil. This could be a problem with the coil itself. They do crap up with precipitant from the heating of the water. You may also have a malfunctioning tempering valve. It regulates the output temp as the coil cannot do so. I think the first culprit is the TV

An old coil will definitely turn cold quickly...they can be cleaned, but they are inefficient...better off installing an indirect fired water heater, connected to your boiler.