I would backup any important data before continuing, but the issue is very likely due to the file system on the external drive. Apple and Windows only share a small number of file systems like exFAT that are fully cross compatible. Macs can read from NTFS and have some capability to write but IIRC it is turned off by default and it can corrupt your drive. A better option is to use free Paragon tool from Seagate to mount an NTFS drive on a Mac.
I have not tried to eject the drive it simply stops being recognized by my iMac. Time Machine sees the various Backups while the drive is working and will try to start a backup but then simply stops seeing the drive after 2-5 Minutes.
There is a tool, that can allow you to read files off a Mac formatted drive using a PC, but if you intend to use it often I would probably just go with exFAT if your Mac is relatively new.
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