Even without Rigify you get junk animation tracks. At least I get in my own hand made armatures. But what I do is keep an eye open in the NLA editor if there is any animation on objects I remove their tracks.
I'm gonna try and explain NLA and also FBX export to UE4 here, at least as my workflow goes.
I always create new action before I start animating to keep the "root" timeline clear. Open Dopesheet Editor, and switch to Action Editor (yellow cube icon). In this case Jump_1_End. Once done click the (F) to make a fake user and then the dubble down arrows (in NLA editor or to the right in header) This will save the Action to a NLA Track (you can rename them, something better than NLATrack.001 etcetera. I did rename my NLA Tracks to my Action Set name.)
Export to FBX, I use the default settings exporting both Action and NLA Tracks as separate Anim Tracks in FBX file.
You can check your export in Autodesk FBX viewer, I do this often to check that export from Blender and import in UE4 is correct. Somehow Unity can read .blend files and get Action animations without pushing down to NLA tracks and no junk data. How they do it, I don't know. I wish UE4 could read .blend files also :)
Anyway here you can see junk anim tracks, try and clean up in Blender by removing animations on objects etcetera and just have NLA track and Armature animations. To keep the junk animations to a minimum.
Drag and drop, or import how you want. But drag-n-drop requires you're hovering a couple of seconds with the FBX file in the Content browser. Once loaded (just went with default settings, it could find the skeleton and all) I clean up and remove the unwanted animation tracks. This works for me, in time I might find a better way. It's not super beautiful, but it gets the game done.
in the NLA Editor in blender you can stack and repeat animations this is good for checking loop points and maybe just have a overview of your animations before exporting. I use this for mock-ups sometimes. You can also blend NLA Tracks when doing longer animations, but in this scope. Game asset creation, you kinda just want to have each Action set in a separate NLA Track layer to get all animations into UE4.
To check your animations on your armature mesh, or skeleton mesh in UE4 you need to make a animation blueprint with simple blueprint logics, there are other tutorials on the internet on this and since it was not your question check unreal engines youtube channel.