The heat that reaches us is in the form of infrared electromagnetic radiation. As such it is not affected appreciably by passage through a vacuum. When it strikes an object that can absorb its wavelength the energy is converted from electromagnetic to heat (not an exact process, but good for a first approximation)

A vacuum is not "cold" in quite the same sense in which cold air or a cold object are cold - a vacuum cannot be warmed up - there just is nothing there to be warmed up.

There is very little to be warmed up on the way from the Sun to the Earth, so the IR rays travel without much trouble until the atmosphere.

And even THROUGH the atmosphere! Air absorbs very little IR light. Air has to get into direct contact with warm objects to get warm, or to mix with warmer air. You can make an experiment: turn on an IR lamp (maybe it works with a normal old-fashioned light bulb) for some time, then turn it off and feel the air and the objects close to the lamp. You will notice that the objects are feelably heated up, while the air isn't (or only close to the objects).

Actually, the Sun heats the surface of the ground first, and then the surface heats the air (quickly) and deeper layers of ground (slowly).

Why is air heated up quickly? Because in the air there also is less stuff (molecules, atoms) to be heated up than in a solid or liquid. The hardest ot heat is water, that's why oceans make a balanced climate.

You are quite correct in saying that heat travels from hot to cold. The thing is, you need a mechanism for that to happen. Think of coffee in a mug. Heat is moves from the hot coffee to the cooler mugby conduction, contact between the coffee and the mug. Heat is moved from the mug via radiation of infra-red light and conduction, contact between the mug and the air. If we put the coffee in a thermos flask, the silvering on the thermos reflects heat back into the coffee and the lack of air in the thermos walls removes the mechanism of conduction.

Light is radiated from the sun as photons. For those photons to transfer their energy to any substance they have to interact with it. Put simply, the photons need to hit an electron to move it into a higher energy state. In a vaccum there are almost no electrons to hit and so there is no effect on the photons. In the atmosphere molecules are not tightly packed and so thee is not much interaction between photons and electrons associated with the molecules. Also, the energy has to be delivered to the electrons in quanta, that is to say that any amount of energy is not going to work, it requires just the right amount of energy in the photon for it to be absorbed. The energy in a photon is a funtion of its wavelength and so only certain wavelengths of light will interact with the molecules in the atmosphere. Once the photons reach the ground things are a bit different, molecules are more tightly packed and the probability of interaction is improved. Even so, some wavelengths of photon are reflected, this is why you can see things and they are different colors.