Peturbative string theory does not derive its S-matrix Feynman expansion from a more primitive Hamiltonian formulation, the S-matrix in Feynman form is the starting point, and the Hamiltonian formalism in the usual time slicing is only valid in the zero-slope/infinite-tension field theory limit. Because of this, the consistency of the perturbative expansion of string theory is established by completely different methods than those of field theory, and it is good to be skeptical. Establishing that the string expansion is consistent was a difficult first step in string theory.
Historically, this took many years, from 1968 to 1981. One important milestone was Scherk's early derivation of point Feynman diagrams from string diagrams, which answers your particular
The string expansion was originally constructed to obey unitarity order by order, so that unitarity is guaranteed by definition once there are no ghost poles propagating in loops. But the result of this construction has more mathematical beauty than the motivation, because the unitary solution is obviously interpretable as a sum over moduli of worldsheets with genus. So to prove unitarity order by order in perturbation theory, you need to prove that summing over world-sheet moduli at each order in the Polyakov expansion reproduces the cut-constructed string amplitudes from unitarity.
The demonstration is by cutting and sewing worldsheets, and this demonstration appears in Polchinski's string theory books, other proofs are hard to find, because unitarity came first historically, and the worldsheet expansion was derived from it later.
Once you have a finite unitary expansion, you should be happy that the perturbation theory is consistent, although this does not demonstrate that the nonperturbative theory is well defined.
String field theory is defined by creation and annihilation operators for each string mode, so that you can make a background of string modes of each type. This gives an intuitive picture of the string excitations as coherently superposing to nonlocal fields which can have a classical real or Grassman value, like ordinary quantum fields.
But establishing the consistency of string field theory generally requires the consistency of ordinary string expansion. The string field formalism is restricted to light-cone gauge, and to certain types of string theory, and favors open strings, unlike the tube-diagram formalism. It doesn't treat branes as equal to strings so it doesn't help with a nonperturbative formulation as much as was hoped in the 1970s. String theory is not field theory of infinitely many nonlocal fields, even though it looked like that originally.
while AdS/CFT is a nonperturbative formulation of strings on the AdS space, it doesn't make ready contact with string worldsheet expansions because it is complicated to describe string theory on a curved background (other than formulating effective quantum field theories and classical equations of motion for the background, which are easy). There are Maldacena's pp-wave backgrounds, for example. But this is the wrong approach (I don't mean Maldacena's pp-wave approach is wrong, I mean the approach of justifying perturbative expansion from AdS/CFT).
Once you know the perturbation expansion for strings is unitary (and finite, and Regge-behaved), there is nothing more that AdS/CFT is going to add to this, because what other consistency condition or well-definedness condition is there?
The split of Feynman diagrams into leg corrections and vertices is only necessary if you are trying to do a renormalization analysis where you have some of the interactions get absorbed into particle mass renormalizations and the rest go into vertex corrections and actual scattering. In the string theory, the final S-matrix is for massless states (or S-matrix like asymptotic field relations), which do not get external leg mass corrections. The particles which become unstable are no longer on external legs.
There is no need to do an infinite renormalization, so the lack of split into leg and vertex is not a problem.