Smoothing current drawn by PWM load
You're talking about two separate issues, the EMI and inrush currents.  EMI filtering won't help much with inrush current.  

As far as inrush, it looks like you need to slow the charging of that honking big (10mF) cap.  There are two common approaches:  


Thermistor This is basically a resistor with a negative temperature coeffcient.  It starts out with high resistance like 10 or 20 Ohms, then as current flows in it, will warm up and end up with resistance of something like 0.2 to 0.5 Ohms.  It is a pretty simple approach.  Problem is it relies on thermal time constants, so if system is operating and is turned off and then turned back on again over a time period which is long for the electrical time constants but short for the thermal time constants (usually 15 to 60 sec or so) the system will see a full inrush.  
Active inrush limiter.  These usually use a P or N channel FET to limit current during start up.  These are made by many manufacturers like Linear Tech, or TI. Here is an example from Linear Tech, the LTC4356 .  There are so many choices, with different features, that it makes no sense to go into it further here.  


For EMI you will need carefully designed filters that, yes as JIm Dearden implies, will involve inductors.  Every subsystem within your system will need an appropriate power and interconnect filter.  All the filters will need to be designed so that they don't have impedance mismatches that could cause oscillation and interactions inside the system.  What that means, and how involved it is, really depends on a lot of interrelated issues that will be system specific.