The factory warranty stays with the mower, and the HRX comes with 4-year warranty. You just need to ask your Honda dealer to update the registration information online with Honda to enter your name and address as the owner of the mower. The warranty covers factory defects, and provided the dealer determines this is the case, it will likely be covered under warranty.
There are known problems with some versions of the AutoChoke; HRR-models can require an updated choke control assembly to cure the binding problems, as well as a revised thermowax cylinder. Both are separate devices from the carburetor itself, and can be replaced while retaining the carburetor. The Honda dealer can make the decision on this upon inspection of the mower. Use this link to find one in your area:
Find A Honda Dealer
While it is technically possible to retrofit a manual choke, it would be costly (different cable, different control lever, different carburetor, etc.) and would also likely be considered "tampering" by the EPA, which has pretty strict rules about any modifications to emissions control parts of an engine, and yes, the choke is considered such a part. Better to get the AutoChoke repaired, because when it is working correctly, it is a nice feature.