Does Quicken import credit card transactions and categorize them?

In a word, yes.

In more words: there are several ways this can happen. First, Quicken by default will try to auto-categorize your transactions. sometimes that works well, sometimes not-so-well. If you have categories of your own you've created, this won't work initially (how would Quicken know what category of yours to use?).

But Quicken has what are called QuickFill Rules for Payees, in which categories (and, optionally, the other fields of a transaction) can be stored an re-used in the future. There are global preferences you can set to do this automatically or not, and a checkbox you can set on any individual transaction to override your general preference in order to tell Quicken whether or not to save a QuickFill Rule. QuickFill Rules can be locked so they aren't changed by future transactions. QuickFill Rules can be created manually as well as automatically from new transactions. So there's a lot of ways to use QuickFill Rules to fit your own preferences for working with Quicken. (Users had lots of different ideas about how QuickFill Rules should work, so the developers eventually added enough preferences and controls that everyone can d it the way they want.) It can take a little time initially setting up/configuring your QuickFill Rules for your frequently-used Payees, bu as you move forward, most transactions should be auto-categorized the way you want.