How can an object absorb so many wavelengths, if their energies must match an energy level transition of an electron?
Your misunderstanding is very common and quite easy to make. Basically, what students are usually introduced to first is the thermodynamics of ideal monoatomic gasses. This is good because it is simple and easy to understand, but can be problematic because features specific to the simple substance can be misunderstood as general features of all substances.
In an ideal monoatomic gas light can interact either by scattering or by absorbing an amount of energy corresponding to an atomic transition*. Note, in the latter case the photon is not absorbed by the electron but by the atom as a whole because the atom has different internal states corresponding to the absorbed energy. As a result ideal monoatomic gasses tend to be transparent except at a few narrow** frequencies.
Now, consider a molecular gas. Just like an atom has internal states that an electron does not, similarly a molecule has internal states that an atom does not. Some states correspond to electron transitions in the molecule, but others correspond to rotational or vibrational modes. The molecular electronic transitions combined with the molecular vibrational and rotational transitions gives rise to a multitude of absorption lines, often forming continuous absorption bands, so many times these are visibly not transparent.
Now, consider a solid. Just like a molecule has states that an atom does not, similarly a solid has states that a molecule does not. The rotational and vibrational modes gain additional degrees of freedom and can act over fairly large groups of molecules (e.g. phonons). These states can have energy levels that are so closely spaced they form continuous bands, and are called energy bands. Any energy in the band will be easily absorbed. This makes most solids opaque as they absorb broad bands of radiation.
Finally, when a photon is absorbed it may be re-emitted at the same wavelength to fall back to the original energy state. However, if there are other energy states available then the energy can be emitted and retained at different energy levels. For example, a UV photon could be absorbed and a visible photon could be emitted along with an increase in a rotational degree of freedom.
*Even for an ideal monoatomic gas there are other less common mechanisms like ionization and deep inelastic scattering, but for clarity these are neglected here.
**Note that even for an ideal monoatomic gas the frequency bands are not infinitely narrow but have some breadth. This is caused by two factors. First, the width of the peaks is fundamentally limited by the time-energy uncertainty relation which says that $2 \Delta T \ \Delta E \ge \hbar$ where $\Delta E$ is the width of the energy band and $\Delta T$ is the lifetime of the transition. Second, random thermal movement of the gas will cause Doppler and pressure broadening of the frequency band.
The other answers cover almost everything, but I would like to add that at any temperature above absolute zero, there is a degree of line broadening caused by Doppler shift: some of the atoms are moving towards you, and others away, and that will mean that in your reference frame they can absorb many different frequencies of light. This is important in astronomy.
Dale and Arpad already gave great answers, but I want to correct something that you said that also contributes to your confustion:
transmission occurs when the energy of an incident photon does not correspond to any electron's energy transition within the material. Therefore, the photon does not interact with the atoms / electrons and is transmitted through.
This statement is not correct. The reality is more close to the statement you gave in reflection:
I believe that a photon is absorbed by an atom, exciting an electron. The electron, however, almost immediately transitions back into a lower energy level, emitting a photon of identical wavelength.
This "momentary absorption" is what causes the refractive index of materials to emerge. The closer the frequency of the photon to the frequency of an energy transition in an atom, "the more time it spends absorbed until re-emitted", this is why the refractive index gets higher the closer you are to an absorption line.
This absorption of energy and re-emission is called Rayleigh scattering, and the re-emitted photon can be emitted in any random direction (with the probability distribution following the antenna radiation distribution). However, because this happens in multiple atoms, the waves constructively interfere only in the forward direction, and destructively interfere in any other direction. This is explained wonderfully by Boyd in his nonlinear optics book:
