Which algorithm is perfect to determine best fit analyst based on multiple factors?
One approach is a set of models which take the ticket as an input and churn out scores for how critical the ticket is, what kind of skill set is required, what kind of satisfaction score will be needed, and the communication skills needed.

Based on these predictions, you could then write a simple rule-based algorithm to determine the optimal analyst to send the ticket to. These rules would be based on the analyst characteristics you mention in the question.

Depending on the form of the ticket input, the models above may require an element of natural language processing (NLP). You may also want to research recommendation systems - it's very likely that something similar has been done before.