How quantitative is the theory of evolution right now?
Can we tell if a species will develop a specific character, based on the environment they're in?


This is a major part of what quantitative genetics does. Or more detailedly, quantitative genetics can predict, for a population, changes in a trait given a selection pressure. First of all, for evolution to occur there must be variation in the trait you are making predictions about. Secondly the accuracy of the scale of the predictions matter greatly for the evolutionary model to be tested (eg, "Will body size increase?" vs "Will incisors increase 3mm in 300 generations?").

Several things have to be known to predict the evolution of a given trait:


The number of genetic sites controlling this trait (QTL, or quantitative trait loci). If the all the loci for a trait are known, the nucleotide heterogeneity within the population of those sites can be quantified. The number of genetic loci are important for estimating how often they will change from new mutations.
The effect of each genetic locus on contributing to variation to the trait (additive genetic effects).
The heritability of the genetics controlling this trait, (does the environment contribute to variation in the trait (low heritability) or do genetic differences cause more variation in the trait).
The strength and consistency of selection  (how fit are individuals in that environment along to phenotypic distribution compared to previous generations). Is this trait being selected in isolation, or are other traits which share the same QTL also under selection.
The effective population size.
The rate of migration between this and other populations.


So yes, given the our knowledge of quantitative and population genetic theory, predicting evolution of a given trait is possible. Just as predicting the future is possible with the laws of physics if you know the position of every atom in the universe. It's important to remember that organisms are constellations of 'traits', so predicting the evolution of all traits which make up the individuals within a species could be similar to predicting the futures of several multiverses.