Q1) When an electron in its ground state absorbs a photon, it absorbs its energy and "jumps" to a higher energy orbital. While it transitions between these orbitals, does it emit EM radiation? If so, of what frequency? If not, what prevents it?
First of all, it is not the electron that absorbs the photon energy, but the whole system nucleus-electrons.
The transition can be regarded in different ways. The simple verbal description that is so often used draws photon as a localized object (a mass-less point) with definite energy and momentum (and sometimes, even position), and the absorption on atoms or scattering on electrons to be a very short, or even instantaneous process. This is an extremely simplified and seriously misleading description of how our equations describe the interaction. This can be called a "buckshot theory of photons", because we have those flying things with definite momentum that get localized in all pictures showing interaction with charged particles. There is no room in this story for "EM field" or "EM radiation" with their well-known and verified properties such as frequency of oscillation in time, or polarization. It is thus just a simple story to tell which recovers some results of the theory.
However, in the best equations we have, EM radiation is described by a field that exists everywhere, and this field evolves continuously with time (at least, most of the time), including during the transition. The transition is a process in time that takes some time, during which the atom is not in "stationary" state, that is, state with well defined energy, but is in superposition of two or more such states. During such superposition, average expected current density in the atom is non-zero, and oscillates in time in a complicated way, as a sum of oscillations at frequencies $f_{mn}$ that are proportional to differences of eigenenergies $E_m,E_n$ of the atom.
Different frequency oscillations of the current are present with differing intensity, related to initial state of the atom and character of the incoming primary EM radiation. If the incoming primary EM radiation frequency is tuned to a transition between two eigenstates, current oscillation at this transition frequency becomes the strongest.
To be consistent with Maxwell's equations, there should be a secondary EM radiation due to electrons, centered at the region where the atomic electrons oscillate. When the system is in a process of absorbing the incoming radiation, the system produces EM waves of the same frequency but almost opposite phase, so the EM radiation in the original direction behind the atom is strongly suppressed (absorption). When the system is in a state of stimulated emission, it radiates with phase that is in-phase with the incoming EM radiation, and thus behind the atom, it boosts the incoming EM radiation in the original direction.
Q2) If an electron absorbs a photon, the electromagnetic energy is transformed into kinetic energy. Can an electron absorb a photon only when bound to a nucleus? Is it right to think that if an electron in empty space absorbs a photon, it causes it to accelerate and this acceleration on a charge simultaneously causes the release of EM radiation (another photon) perhaps of lower frequency?
Electron does not absorb the photon, the whole atom does. And this energy does not all turn into kinetic energy, but some of it turns into potential energy (because the whole system gets into excited state with higher expected average of potential energy).
When the electron is alone, not a part of any atom or other composite system, it can only scatter the incoming radiation, meaning that while the electron is gaining some energy (and change of momentum), it produces also outgoing EM radiation with some energy and momentum leaving. This experimentally implied fact can be explained by the assumption that electron cannot, during interaction with EM radiation, easily change its rest mass. A massive body completely absorbing another body with energy and momentum (such as a mass-less packet of radiation) would lead to increase of mass of the massive body. If increase and decrease of mass does not happen, then the interaction process must result in some energy and momentum leaving away from the electron.
Thus electron in empty space cannot, as far as we know, absorb a photon, it just does not happen. If it happened, the electron would increase its mass, but we do not observe heavier electrons resulting from scattering of EM radiation off the electrons.
But indeed, the scattering process near single electron, where radiation transfers some energy and momentum to the electron, and another outgoing radiation with somewhat different characteristics is produced by the electron, can be understood in classical theory as the electron being accelerated by the primary radiation, and thus radiating its own secondary radiation. This secondary radiation can have variety of different directions, intensity then depending on this direction in the expected way (e.g. in case of linearly polarized EM radiation, the strongest scattering will be in the plane perpendicular to electric field of the primary wave). The scattered radiation can have shifted frequency depending on direction, if the electron moves in the lab frame; this would be an example of the Doppler effect.