On a check valve... is cracking pressure something to take into consideration when adding up pressure drops?

Not exactly. The cracking pressure is the pressure that the check valve needs to open (to permit flow) the flow rate through the vale is related to the pressure drop. If you haven't reached the required cracking pressure you won't have anything flowing through the valve.

It's my understanding that if you relate this to a mechanical system, such as a cart, the cracking pressure would be the same as the static friction. They resist up until the point where the system starts moving. The pressure drop across the valve, once it has opened, would be the equivalent to the kinetic friction of the cart.

You may want to read up on sizing check valves. Check valves that are improperly sized can do all kinds of bad things to themselves. For instance a check valve that is too large will "chatter" and can become damaged.

And it really depends on the design of the particular valve, but for most check valves the pressure drop across the valve will always be greater than the cracking pressure. So the most probable answer to the original question is yes, the cracking pressure is added to the piping pressure losses.

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