I will use plaintext as an example...
A plain text file does not contain any meta data, i.e. it is only an octet stream without structure. The only thing meta on this octet stream is its size. But, if stored on a UNIX file system there are several meta data associated with this octet stream, like the file name, modification time, owner, group, permissions, maybe ACL's etc.
... I transfer it to another computer via a flash drive, is the inode for the file copied?
The inode itself is specific to the instance of the underlying file system. It will not be copied over to the flash drive. But some of the information contained in the inode might be, like the modification time. But the details depend on the capabilities of the underlying file system of the target, i.e. different file systems support different kind of information in the inode (or similar data structures). For example a (V)FAT file system which is usually used of flash drives does not support UNIX file permissions. It also depends on the way the copying was done, i.e. simply calling
cp will not copy over the modification time but set it to the current time in the target. And even if the target system supports UNIX file permissions and owner, the original owner cannot be copied over unless the user doing the copy has the permissions to set the owner (i.e. is root).
... But are there any types of metadata separate from the inode?
The file name itself and the directory are not contained in the inode. In UNIX you could even have different file names or some file names within different directories point to the same inode, which essentially means that some same data can be accessed from different path.
What about in the case of an image file?
There are a variety of file formats for images with different capabilities. For example JPEG files can contain EXIF, IPTC or XMP blocks which are contains for a variety of meta data, like creation time of the photo, camera used, location where the photo was taken, copyright information and much much more. Since these information are contained in the octet stream of the image file these will be implicitly copied too when the file is copied.