We study several classes of symbolic weighted formalisms: automata (swA), transducers (swT) and visibly pushdown extensions (swVPA, swVPT). They combine the respective extensions of their symbolic and weighted counterparts, allowing a quantitative evaluation of words over a large or infinite input alphabet. We present properties of closure by composition, the computation of transducer-defined distances between nested words and languages, as well as a PTIME 1-best search algorithm for swVPA. These results are applied to solve in PTIME a variant of parsing over infinite alphabets. We illustrate this approach with a motivating use case in automated music transcription.