I'm far from being an expert on BBD and related algebraic geometry, but the question about "construction" of irreducible representations in the infinite dimensional context has to be approached with great care. Although direct constructions are possible in a few special cases, the main problems for Lie groups or their Lie algebras usually require an indirect approach.
For example, in the BGG category (say with integral weights) there is an easy construction of Verma modules using induction methods; the formal character is also easy to exhibit. But the unique simple quotient cannot in general be constructed even by sophisticated methods. Instead you try to imitate the BGG approach to the finite dimensional character formulas: in effect, express the unknown formal character as a $\mathbb{Z}$-linear combination of the known Verma characters. This can provide an effective algorithm based on the partial ordering of weights. The integral coefficients here still need to be found. Kazhdan-Lusztig predicted these in terms of recursively computable polynomial values at 1, but so far only the geometric methods of Beilinson-Bernstein or Brylinski-Kashiwara have been able to prove this.
Similar predictions are made by Lusztig in prime characteristic for representations of semisimple algebraic groups, but only proved for large primes (and with no definite prediction for primes less than the Coxeter number). Analogues for quantum enveloping algebras at a root of unity have by now been attacked successfully using the characteristic 0 geometric methods. In the Lie group direction, Vogan and others have pressed further with partial success in the spirit of the KL Conjecture. But all of these problems are extremely difficult. At the end of the day, very few concrete constructions of irreducible representations are found. Usually one settles for some kind of "character" information. While the classical Borel-Weil theorem inspires some of the later moves, the story gets much more complicated.
Concerning the added "further question", my remarks apply equally well to affine Lie algebras in most cases. But the situation there for the critical level or for the somewhat parallel finite dimensional modular theory mentioned above is less settled. There has been a lot of recent progress in both cases, for example in the modular theory by work of Bezrukavnikov, Mirkovic, Rumynin. In all cases, the results obtained by localization or other geometric methods lead mainly to multiplicity formulas and recursive computation of characters rather than explicit construction of simple modules. But in finite dimensional cases, even their dimensions have been elusive.