Best clustering algorithm to identify clusters and determine the closet cluster each individual response is near?
Since they are categorical variables, I would cluster them using the k-medoids clustering method. Before applying this method, one-hot encode all the predictors.

See a tutorial here:
https://towardsdatascience.com/k-medoids-clustering-on-iris-data-set-1931bf781e05

Sklearn has an implementation:
https://scikit-learn-extra.readthedocs.io/en/latest/generated/sklearn_extra.cluster.KMedoids.html