Wrong:
"since infrared waves have a shorter wavelength"
Infrared has longer wavelength than visible and visible longer wavelength than ultraviolet .
White is a term for visible light mixed wavelengths. In the plot you can see that almost half of the sun's radiated energy arrives as visible light. The white buildings reflect this visible light which otherwise,impinging on the surfaces would be absorbed and turned into infrared by the interactions, adding to the arriving infrared.
What is absorbed and what is reflected depends on the chemical bonds of the surfaces, whether the incoming radiation can excite molecular states of the materials. Infrared is in frequencies/wavelengths of the black body radiation of bodies in the temperature ranges comfortable for the human body, so they easily raise the vibrational and rotational levels of solids and liquids and the kinetic energy of gasses.
37C curve seen here practically all in infrared, and lower temperatures more so.
Thus white paint will not reflect infrared as efficiently as visible, a large part of infrared will be absorbed as also some part of visible will scatter at the surface and degrade to infrared. Infrared can be reflected by metal mirrors, from the collective fields in metals . If you put aluminum foil in front of a heater you are sheltered from most of the heat which is reflected, but some of it is absorbed as can be seen by touching the foil.
If I expose an object to EM radiation only from the infrared spectrum, will it only reflect back infrared?
Yes, but most of it will be absorbed ( except by mirror metal surfaces) because the materials have the receptors for these wavelengths. This is due to the fact that larger wavelengths have photons with less energy which cannot excite higher energy levels.The energy of the photons goes as h*c/lamda where h is plancks constant, lamda the wavelength and c the velocity of light.
Is this true for other types of EM radiation?
No.Visible and ultraviolet by scatterings degrade their energy down to infrared frequencies, depending on the material.
Is it possible to make an object that looks white and absorbs a lot of infrared radiation?
Usually most of the infrared will be absorbed except by mirror metal surfaces.
If an object reflects most of the EM radiation that it receives of a particular wavelength λ, will it also reflect most of the radiation it receives of wavelengths less than λ (and absorb most of the radiation of wavelengths larger than λ)? Is this why objects that reflect most visible light (and hence look white) also reflect most infrared radiation (since infrared waves have a shorter wavelength)?
There is no such rule. It depends on the material and its chemical bonds.
The term "light" is a little ambiguous, because for some it means visible light, and for others it means any form of electromagnetic radiation. But I agree with you, that black and white should be more about visible light than infrared radiation. If we are right, then it's better to wear white, because white will reflect the incident visible sunlight, while not doing anything different with the infrared radiation from our bodies. I suspect the conventional wisdom may have got this one right, and the "correction" to it might be in err. Perhaps it needs to be an episode on "Mythbusters."
Best Answer
The tricky bit is that there are other effects happening in stars that lead to the spectrum being "incomplete". Quoting the last paragraph of the link mentioned by OP,
Light will leave the surface of the Sun at pretty much a black body spectrum, but the Sun contains multitudes of hydrogen and helium, for example. These elements (just like any other) have quite specific spectral lines: the differences in energies between their energy levels are quite specific and they can absorb only photons with those precise energies. Due to this effect, the frequencies corresponding to these precise energies will end up deprecated on the spectrum we can detect on Earth. Essentially, when we look at sunlight, we see there are a few frequencies missing.
These effects were first noticed, if I recall correctly, by Fraunhofer. By cataloguing the spectral lines of different elements, one becomes able to identify the composition of bodies by looking at their spectra.
If we had a different star, the spectrum could be different, since it depends on the specific chemical composition of the star's atmosphere.