Can you play sims 4 with an external hard drive?

Yeah it would load files fine off an external drive. Especially if it was trhough an eSATA port, which is the same as being directly plugged into a SATA port and you notice zero difference in latency or read/write speed.  USB 3 is a bit more sluggish in writes but in reads its not that bad, especially if its an SSD you'd barely notice much difference.

Some possibly false info here about the documents folder.  You can actually tell that to be saved somewhere else.  You just right click on Documents folder and click properties, then add a folder to your external drive.  Then "Set Save Location"

This in theory should make every single game that would save to the Documents folder then be saved to wherever the new folder is....  You could predict some problems doing this since some games, and especially older ones wouldn't go by this setting and may manually still try to save it to the Documents folder on C:\ or wherever you have windows installed.

Personally I wouldn't do this, I would keep it in its default location and just back up the folder every so often.  But if you were totally empty on C drive for space and used a lot of big mods, then maybe try this.  Also may be useful if you wanted to store everything, saves and the actual game in one drive so you could transport it to different PC's more easily but each new PC would have to have its documents folder re-directed...

but Backup the entire documents at least twice first though before you do anything.