1) Note that it is considered insecure to have the .htpasswd file below the server root.
2) The docs say this about relative paths, so it looks you're out of luck:
File-path is the path to the user file. If it is not absolute (i.e., if it doesn't begin with a slash), it is treated as relative to the ServerRoot.
3) While the answers recommending the use of environment variables work perfectly fine, I would prefer to put a placeholder in the .htaccess file, or have different versions in my codebase, and have the deployment process set it all up (i. e. replace placeholders or rename / move the appropriate file).
On Java projects, I use Maven to do this type of work, on, say, PHP projects, I like to have a build.sh and / or install.sh shell script that tunes the deployed files to their environment. This decouples your codebase from the specifics of its target environment (i. e. its environment variables and configuration parameters). In general, the application should adapt to the environment, if you do it the other way around, you might run into problems once the environment also has to cater for different applications, or for completely unrelated, system-specific requirements.