If an atom builds up from an ion capturing an electron which state of the atom is more probable as a first transition (immediate after capturing) - with higher energy (excited state) or lower (more close to nucleus - e.g. ground state or some higher than ground). What is the reason and is there a mechanism? Also is it possible by some conditions to prohibit higher levels (maybe a tuned appropriately cavity) and the recombination to take place only and directly on the ground state?
I suppose you surely mean will it jump from the initial free state directly to a low-energy bound state or go there gradually through higher-energy states.
I would say if the electron initially have high angular momentum, and if the atom is isolated and can only de-excite by emitting photons, because usually only lower multipole emission of photons happens (as others are much slower), the atom can only go to the lowest-energy state compatible with the initial angular momentum, and then shed angular momentum and go to lower energies gradually by emitting series of photons.
So my views is, it depends on the initial angular momentum. It will quickly go to the lowest energy state with angular momentum one or two $\hbar$ away from the initially free state. Whether this state has a high or low energy depends on atom's energy-level structure as well as the initial angular momentum.
which state of the atom is more probable
It depends on the state of the electron and atom being far apart and their relative motion. The process of electron capturing should satisfy the following conditions:
energy conservation: kinetic energy of electron + total energy of ion = total energy of atom after capturing.
wave function overlap (electron and ion should be located in the close vicinity for capture to happen). The excited states of ion/atom are more spread in the space - they will overlap with incoming electron wave function first.
These two conditions make me believe that the atom will be in a highly exited state after capturing the electron. So, we should observe the relaxation process with emitting photons afterward.
I think such electron capturing can be seen as an inelastic scattering process in a way - the free-electron loses all its energy and seize to exist - electron and ion annihilate and an excited atom is created.
An ion is built up by losing the outermost and least bound electron. If there is an electron in the vicinity afterwards, it is placed at this outermost position.
Only if an electron with high kinetic energy hits the ion is it possible for electrons further inwards to be displaced.
In both cases, photons are emitted.