model selection in clustering
To answer your initial question, yes you can use silhouette score with different clustering methods. You could also use the Davies-Bouldin Index or the Dunn Index.
Regarding over-fitting, (this is my personal suggestion) but you could train the model n times on different types of the same data to see if there clustering is the same even though the values are changed. Short example: If you have to cluster 5 apples and 6 oranges, the cluster should be the same for 10 apples and 12 oranges. You can find a bit more detail on this here: https://datascience.stackexchange.com/a/20292/103857
For your third query:
Calculate distances between data points, as appropriate to your problem.
Then plot your data points in two dimensions instead of fifteen, preserving distances as far as possible. This is probably the key aspect of your question. Read up on multidimensional scaling (MDS) for this. Finally, color your points according to cluster membership.
(source for third query: https://stats.stackexchange.com/a/173823)
Regarding pca, its subjective. PCA works well with high correlation. If your dimensions are like apples and oranges then your directly effecting your models performance, so do keep that in check. A bit of eda would help before you dive into that.