Scenario 1: Will an atom absorb an electron with kinetic energy greater than the difference of the initial and final energy levels or must it absorb an electron with kinetic energy similar to that of the energy difference only?
Let us talk of free atoms, gas.
If the atom is ionized, there will be an available energy level that an electron could occupy. A free floating electron at rest relatively to the atom can fall on that energy level and release a photon. In the case of an ionized hydrogen atom ( called a proton),
it will release a photon of energy 13.6 eV .
If the electron is not at rest with the nucleus, the probability of capture is very low, though computable, the excess energy released in the interaction as a photon carrying away the difference and bringing it at rest so as to be captured. The probability is low because extra electromagnetic vertices will be needed to compute the interaction crossection.
So the answer is that predominantly the electron must be at rest to be captured.
Scenario 2: I've read the following statement in my textbook, but I find it hard to believe since we were told that an atom can absorb a photon with the exact amount of energy to the energy difference. Statement: "A photon can be absorbed and cause ionisation if its energy is greater than or equal to the difference between the ionisation level and the ground state, although excitation requires photons with specific energies."
I read this as : in a neutral atom where the electron is bound , as with the hydrogen example above, in the ground state , the electron can be kicked out of the ground state if the energy is equal or larger than 13.6 eV. The electron will carry the balance of the energy of the photon.
Excitation means that the electron is hanging around in the higher than ground state levels, it is still bound by the potential of the nucleus. Ionization is where the electron becomes free of the potential of the nucleus, and since over the 13.6 eV in the example above, there exists a contiuum, it can carry all the left over energy as kinetic energy from the kick.