Control 'start' of an animation by sliders (for mogrt)

Another option would be to Pre-compose your animated layers, moving all attributes and trimming the layers to comp length. Because you have a slider for Fade Time, I would apply this expression to Opacity after you have pre-composed the layers:

t = time - inPoint;
et = comp("Comp 1").layer("Controller").effect("Fade Time")("Slider")* thisComp.frameDuration;
linear(t, 0, et, 0, 100);

Comp 1 is your master comp.

Then go to the first Pre-comp, add Time Remapping (Alt/Option + Ctrl/Cmnd + t), and apply this expression to Time Remapping:

t = time - inPoint;
st = thisComp.layer("Controller").effect("Line 1 Start Frame")("Slider")*thisComp.frameDuration;
t - st

I changed the Name of the controller to Line 1 Start Frame because multiplying by the frame duration gives you a counter that measures frames. 

You can also use valueAtTime(), but if you have a bunch of properties that you want to move down the timeline, Time Remapping is going to be a lot more efficient. 

The only hangup with any of these solutions is that any ending transition, like a fade out, even if the area is protected, could get fouled up. I have not tried it yet because most of my MOTRTS use the layer's inPoint and outPoint and a linear or ease interpolation for movement instead of keyframes so that timing does not change when the graphic is stretched in Premiere Pro.

I uploaded the modified file for you to look at.