Can an electron jump to a higher energy level if the energy is insufficient or exceeds the $\Delta E$? Let's say we have an atom of hydrogen. It has one electron on $E_1 = -13.6~\mathrm{ eV} ~~(E_2 = -3.4~\mathrm{eV})$ energy level. I know that if we fire a photon with 10.2 eV energy the hydrogen atom will absorb it and the electron will jump on the next energy level E2.
And below are my questions.

*

*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?

*Q2: Same question as the above one in this case our photon has
slightly more energy, let's 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?

I have done some research about it and got really confused. Some say that it needs the exact amount of energy ($\Delta E= E_2 - E_1$ in our case $\Delta E$ equals to 10.2 eV) to jump onto the higher energy level some say that it can jump if the energy exceeds the $\Delta E.$ What I really could not find is what happens with the extra amount of energy or maybe electron can be on $E_2$ energy level with slightly more/less energy.
Eventually I want to understand the concept of the reflection. How we see the objects, why they are transparent or glossy or red or whatever else. But this is out of scope of my question.
I'm not an expert though; so mark down the mistakes above if there are any.
 A: 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.
A: If your photon has not enough energy to excite the electron then it will just not be absorbed and will pass by, and if you have an electron with an excess energy, it will be absorbed and a photon with the excess energy will be automatically emitted and the electron will jump to an excited state. So yeah in your case, you might have a photon with $0.1 \space \mathrm{eV}$ of energy which corresponds to a photo with a wavelength of $12.4 \space \mathrm{\mu m}$ so in the infra-red.
