Is there any way to copy an effect from one layer and then paste it into another layer, even if they are in separate compositions? I just don't want to have to re-apply the effect and then re-enter all the same variables every time I need to use it, in exactly the same way.
Select the effect in the Effects Control Panel or in the timeline. Copy. Select your target layer in any comp and paste.
The other alternative is to create an animation preset. I have more than 200 in my library that I created for tasks that I use all the time.
Look into animation presets in the help. They sound like exactly what you need.
Although, you can copy and paste effects too. You select them, edit>copy select the new layer edit>paste
And, if you have CC or CC 2014, you can copy and paste effects with links back to the original so you can change it and the others will change.