Can an electron jump to a higher energy level if the energy is insufficient or exceeds the $\Delta E$?

When a photon hits a boundary condition , three things can happen: a) it can scatter elastically, which means it retains its frequency but changes angle, b)it can scatter inelastically, which means it changes frequency, or c) it can be absorbed raising the energy level of an electron ( in a lattice, in a molecule, in an atom) and a different photon is emitted and phases are lost.

Q1: If a photon with 10.1 eV energy (insufficient to excite electron) would hit the atom of the hydrogen what would happen? Will the photon be absorbed by the atom and immediately emitted and the emitted photon (or photons?) will have the same 10.1 eV energy? Or the photon will pass through the atom or what would happen?

The hydrogen atom hit with a photon of energy lower than an energy level transition falls under a) or b) The photon will scatter elastically in the center of mass with the total atom and go on its way at adifferent angle, or inelastically giving kinetic energy to the whole atom and changing frequency.

Q2: Same question as the above one in this case our photon has a slightly more energy lets say it has 10.3 eV. Again what would happen? Will the atom absorb the photon and excite the electron but since the energy of the photon exceeds the required energy to excite the electron will the atom emit a photon with 0.1 eV energy or what will happen in this case?

If the extra energy of the photon is not within the energy width of the hydrogen energy level, again it will go on its way scattering elastically or inelastically in the center of mass "photon atom" . If the energy of the photon is higher than the ionization energy of the atom, the work function, the electron may be kicked off and the ion proton remain. The photoelectric effect.

One has to realized that at the quantum mechanical level it is probabilities that are important. The probability for a photon of the correct energy difference to raise the electron of an atom is very high, with the wrong energy difference. very very small.

For bulk matter interaction see this answer of mine here.