Newton's law is for convection of heat into the air. The cooling of the earth happens by radiation into vacuum, so it's actually much MORE temperature-sensitive: the heat radiated is proportional to the fourth power of the temperature.
Obviously solar input dominate over whatever energy we add through burning fossil fuels. Nobody contests that. You must have heard of the greenhouse effect. THAT is the issue. By changing the composition of the atmosphere, a fractionally greater percentage of the astounding amount of energy is reflected back to the earth instead of escaping into the atmosphere, raising the temperature of the atmosphere and oceans.
Imagine sitting in the sun on a warm but not hot day. Now imagine you are sitting in a car in the same spot in the same conditions with the windows rolled up. There is no extra energy being released, just the plain old sun, but the temperature you experience is going to be much higher. Indeed, the temperature increase doesn't grow without bound, but it is meaningfully higher.
Right. When we generate electricity via solar or wind, the heat involved has (essentially) been shifted from elsewhere.
That energy was going to reach our planet either way. Whether we use a solar panel to shift it somewhere we can use it, or if it just heats the whatever it would have otherwise ran into, we aren't adding energy to the system by shifting it around.
The waste heat from solar panels is just the energy from the sun, less whatever gets turned into electricity. No new energy was created - if there was, you could just create a perpetual motion machine to solve the problem.
Fossil fuel is stored chemical energy. Burning it releases the energy. Photovoltaic solar power is converting one type of energy (light) into another (electricity), with the waste heat being the same waste heat the light would've created by shining on something else. Solar thermal energy is using the heat from the sun to do work (which may include heating oils, air, or water to drive a turbine to generate electricity) or to generate electricity (using, say, thermal-piezo conversion).
The only effective ways we have to store solar power are batteries, supercapacitors (once they're viable), large heat masses (such as stone or water), growing plants, or putting energy into chemicals for later storage (electrolysis and storing the hydrogen, forming artificial hydrocarbons, recycling aluminum for aluminum/air power cells, pumping water uphill to power a turbine later, pressurizing gases to later drive a turbine, stuff like that).
Not all of the things I mentioned work at every scale. Some of them are more efficient than others. They may all have advantages and drawbacks.
As long as we still have cold seasons, it is safe to assume that solar inputs are dominating our own.