How does the construction of a decision tree differ for different optimization metrics?
Splitting is the same your are optimising for the parameters.

In other words, gini, entropy etc logic will stay the same but the breadth, width, number of features etc will be different when you optimise it with different loss functions.

Accuracy, precision, F-scores etc. are evaluation metrics computed from binary outcomes and binary predictions. They are NOT (but there are modification) for them to be a loss function.
For model training, you need a function that compares a continuous score (your model output) with a binary outcome - like cross-entropy. Ideally, this is calibrated such that it is minimised if the predicted mean matches the population mean (given covariates)