Is there a way to duplicate and flip an animation to the opposite side of the screen?

Without knowing teh full details of the project, this sounds like a job for a nested composition. A nested composition is basically a comp inside another comp. One of the things it allows you to do is create and reuse animation. 

1- Select the layer or layers that you want to convert into a nested comp.

2 - Right–click and choose Pre-compose... from the menu. You could also choose Layer > Pre-compose... or use the keyboard shortcut Cmd–Shift–C (Mac) or Ctrl–Shift–C (Win).

3 - In teh dialog box, give your new comp a name and if you have a choice of radio buttons choose the second one, Move All Attributes... and click OK.

4 - Your layer or layers have been converted into a new composition which is now also present in your Project panel. Just drag a copy out of the project panel or duplicate your layer and you can have an exact copy of teh animation you created.

To flip a layer you can use the Transform menu, Layer > Transform > Flip Horizontal.

You can also scale, move, rotate the comp layer like any other layer.

To change the animation you can double–click to open the comp and make the changes there. Note every use of the comp is an instance of the original so changes to the original comp will be reflected on every instance you made.