Is it possible to design a system where you have one Arduino connected to a DB9 adapter (and a serial device), and have the Arduino take the commands, send them over Wifi (or cellular) to another Arduino that will receive the data and output it to another DB9 adapter (and device), and it's like the two devices are physically connected to each other?
Yes, you can do a "virtual serial link" in many ways, doesn't much matter what the bit in the middle is.
One thing that is sometimes a problem is timing, if one of your devices needs feedback in a certain time than the delay in such a link causes issues.
Yes, but if it really works as expected depends on how fast the link has to be, concerning latency.
If the distance is only some meters, you might take a look at HC-05 Bluetooth modules.
Bluetooth modules HC-05/HC-06 will do this job.
You may want to consider an XBee. They operate exactly like you want.
Their Series 1 is simple to use and can be used to create a virtual wire between devices.