The answers largely depend upon what migration tool you plan to use. I recommend asking in the chosen tool’s dedicated support channel.
In choosing a migration tool, look for one that specifically supports merging two sites. As you know, ensuring there are no overwrites is an issue unique to merging sites that is irrelevant when simply migrating a single site. Not only must overwrites be avoided, but the various ID changes need to be coordinated to maintain the record relationships between tables.
If you’re referring to the export/import tool included in WordPress, see: https://learn.ww.wp.xz.cn/tutorial/tools-import-and-export/
Ok, so the import/export tool link didn’t include the information I was looking for. It only tackled importing into a new installation, not an existing installation. That’s the part I really need to get the details on. In this situation, is it not recommended to use the default wordpress import/export functionality and to use a separate program and/or plugin?
do categories carry over?
Yes, as described in the tutorial.
Will they overwrite existing ones (if the same name)?
Based on my experience with the importer, yes. To be more precise: records with the same slug or ID that already exist will be updated with the new data.
Will scheduled posts remain scheduled or will they have to be rescheduled?
Scheduling is based on the post status. In my experience, it is carried over exactly as is.
Can you only import/export pages and posts or can you import/export media?
The importer includes an option to download attachments as well. You’ll need to enable this option if you want to do so: https://developer.ww.wp.xz.cn/advanced-administration/wordpress/import/#wordpress
If you’re unsure about the process, I’d recommend testing it first in a project copy, such as a staging environment. If something doesn’t work the way you want, you can either check if the importer’s behavior can be customized using a hook (see the plugin page for details) or use one of the many other plugins available for this purpose: https://ww.wp.xz.cn/plugins/tags/importer/