AC shuts but fan keeps running

Your Rh line is Heating, RC is for Cooling , R is similar but systems might be separated and use dedicated lines.

What it sounds like is that you are hooking up your thermostat for the first time ??

Either your wiring is incorrect or there is a program mode where the fan runs. Since your question says HVAC - I am assuming AC .. however if the Unit is calling for heat Rh .. your system will have a cool down period - the Blower will run after the elements have turned off for about 5 to 10 minutes I think.

A typical color coding of wires is this : BUT DON'T BE FOOLED - Not everyone installs things correctly .. So, this is what you might expect to see in terms of colored wires and what they mean:

White – The white wire is what connects to the (auxiliary) heat on your system.
Yellow – The yellow wire connects to your compressor.
Green – The green wire connects to the fan.
Orange – This wire connects to your heat pump (if you have one).
Red – Now, there can be two separate wires for this. 
      One is for cool and the other is for heat, 
      hence the abbreviation “RH” and “RC”.
Blue – This is your common or “C” wire.

So it would be nice to see a picture of your connections in the NEST and at the Air Handler. Your wiring seems to be incorrect. especially when there is a message about Rh and you are trying to cool.

Look Here http://www.thermostatcenter.com/thermostat-wiring/

Also Here : On a thermostat, are "R" and "Rh" terminals the same thing?

One of Nest's selling points is something called "Airwave," which keeps the fan running after the compressor turns off in order to circulate the cool air in the chilled ducts throughout the house. You can turn this feature off in the settings.

Some Aircond model have this function to ensure no condensation occurs on the cooling fins. Without the air flow, water condensation will occur quickly on the cooling fins. Wetness in the fins will cause bacterial growth. With the fan running for sometimes, all moist will dry off and the cooling coil will return to room temperature quickly.

With this system in place your Aircond won't produce weird odour the next time you switch it on.

Have you checked to make sure you don't have a hole in your ductwork that is pulling moist air into your home? If you A/C is a "package unit" especially, you must determine the integrity of the ductwork, especially the large diameter hoses that connect the unit to the house. If there's a breach, you could be pulling in too much humid air from outside which can cause the evaporator coils to freeze up. Many units sense that the coils are still cold and will automatically run the fan until the temperature comes up. In this case, the evaporator fan will tend to cycle on and off or simply run continuously in an attempt to disperse the cold air.