Will Docking Station make Early 2013 MacBook Pro faster
Would using a docking station keep it cooler and make it appreciably
  faster?


No, a docking station has no "compute" power built into it so it can't offload any processing of any kind from the CPU or GPU.  This means it won't be cooler nor faster since the CPU and GPU are still handling the workload.

The reason you are seeing a "slowdown" when your 4K monitor is connected is because you are driving an external high resolution display with a integrated graphics chipset - the Intel HD Graphics 4000.  Being that it's integrated, it allocates up to 1.5GB of system RAM and uses CPU cycles for GPU processing.

Your MacBook Pro is slower because you are pushing it to (and past) it's limits in terms of capabilities.