does Altalix need to run Stellar servers?
No, you don't need to deploy any servers to become an anchor. "Anchor" is rather a designation of a trusted entity concept, it does not imply that you need to meet some specific requirements. What you described falls under the description of an anchor:
Anchors are entities that people trust to hold their deposits and issue credits into the Stellar network for those deposits. All money transactions in the Stellar network (except lumens) occur in the form of credit issued by anchors, so anchors act as a bridge between existing currencies and the Stellar network.
As for the second part of your question...
Or can we just call APIs on existing Bridge/Federation/Compliance servers?
I'm not aware of any publicly available Compliance or Bridge servers, therefore I believe that you have to run your own instance if you need this functionality (it have to be tightly integrated with your internal accounts database).
You also may want to use some advanced features provided by self-hosted Bridge and Horizon servers if you are aiming to maintain high transactions volumes. For example, Stellar-hosted Horizon instances has request rate limiting, Bridge server provides non-blocking interface for sending multiple transactions etc.