How to copy all properties layer or composition ?

My first inclination for projects like this is to open the Extended Graphics workspace and create a MOGRT for the animated composition. You can create editable text fields that have animations, change text color, change alignment and position, and then export the MOGRT and open it in the Graphics workspace in Premiere Pro. You will never have to go to After Effects again to change the text or even the duration of the animation. 

If you must stick with After Effects, then I suggest that you create and save animation presets for all of your animated layers. Then all you have to do is apply them to new layers to have all the modified properties that you saved applied to that layer. 

If you are new to After Effects, I suggest you spend some time here learning the basics. Create Motion Graphics templates with Essential Graphics panel will show you the basics of creating MOGRTs. 

Let me know if you need more help.

You can write expressions that look at the timing of in and out points, layer markers, and comp markers, then use that data to sync or drive animations. If we knew exactly what you were trying to accomplish or even what properties you were trying to mimic, we could point you in the right direction. 

You cannot copy the trim or markers. That would require scripts. Check AEScripts.com. Otherwise you copy property values and keyframes by actually revealing them ansd selecting them, then revealing the same properties on the other layers and pasting them there. You may want to rwead the online help on some of these basics.

Mylenium