external hard drive ejected repeatedly

Well of course check your cables to ensure they have good connections. Please restart in Safe Mode ( Safe Mode Reset ) with all external wired peripherals disconnected. Then shut down and connect your EHD and restart normally and test.

Unless your Seagate is old, it should be okay for 3-5 years. As a rule of thumb it is good practice to exchange HD's that are 5 years old (or older) with new drives because HD's often give you no warning when they are about to die.

Your strategy of using 1 EHD to use as a Time Machine backup and storage is shortsighted and a 100% guarantee you will lose your libraries stored on it. The reason being is ALL hard drives fail, it is not a matter of if it is a matter of when. The means when it fails you will lose your libraries and all of the back up!!!!!!!!!!!

External HD's are relatively inexpensive. I would first recommend buying a higher quality EHD than a Seagate, Toshiba or WD. These types of drives use very cheap enclosures that tend to overheat and die. What many experienced users use are the OWC Mercury Elite Pro due to their reasonable pricing, great build quality, and excellent support. I have some that are 15 years old that still work, yes the original HD's have died but you can easily fit a new HD inside in less than 5 minutes. With the type of drive you have it is a throw away which will end up in a landfill somewhere.

Please get yourself 2 drives, one for your Time Machine backup and another for your media libraries.

Some have reported this issue and others have not. I have seen one of my drives do this about once a month which is a USB drive. I have another drive that is TB drive and never happens. Both drives are the same but in different enclosures (USB & TB). I have not seen any definitive answer by anyone what causes this.

Your welcome and let us know if the Safe Mode restart helps.