Yes, the steps would be the following:
Add or change the Alternate Access Mapping
Edit the IIS binding for the IIS Site hosting the Web Application
Update the Search content source with the new Web Application URL and perform a full crawl
If you want to move the SharePoint Servers to a new Active Directory domain, yes you need to create a new ...
Unfortunately there's no direct way to migrate OOTB from on-premise to Cloud. And the 2003 version is too far away.
According to my research, the avepoint you mentioned really doesn't meet your needs, it only supports SharePoint 2007 at most.
You can try some other tools, such as josh's recommended.
In addition, you can refer to this:
ShareGate says that they support 2003 as a source but not as a destination, so that should work.