Can one proton attract two electrons? Suppose that in an empty space there is only one proton. This proton would have created a field of positive charge which should attract possible negative electrons, so now we add two electrons on opposite sides of the proton in a similar distance. What happens then?
Are both electrons attracted to proton with full/half the force before the one that was a little bit closer reaches the proton first and forms a bond and the other one now glides past neutral hydrogen atom?
 A: What you are talking about it exactly what is known as the protide(hydride) ion. A protide ion has a negative charge - one proton and two electrons. In another case, any covalently (even datively) bonded hydrogen atom has two electrons, generally one of itself and one of the atom bonded to it, though in the case of a dative bond (such as in ammonium ion) both of the electrons are supplied by the other atom.
Though the protium/hydrogen atom is neutral at zero net charge, it has internal structure and therefore there are always parts of it left "exposed" to allow it to attract more electrons. The number of electrons it can attract is then the matter of orbitals. 1s orbital which a hydrogen atom has can hold up to two electrons, thanks to electrons' half integer spin that allow them to bear opposite spins so that Pauli's Exclusion Principle doesn't make them repel too strongly. Any more electrons will repel too strongly and will need to occupy an orbital of a higher energy level, and the energy level might be too high that it'd rather be a free electron than to bind to the hydrogen nucleus.
A: A proton and two electrons is a quantum mechanical problem. The hydrogen atom has energy levels which can be occupied by one electron if its energy is such that it can be captured, but a second electron can also be captured again depending on its energy. The energy levels will be modified but a solution exists.
The negatively charged hydrogen atom is an anion.
