1- As illustrated in Image 02, Am I missing any resistors?
Not resistors but the wiring is wrong. You're using the hardware serial interface but wired for a software emulation (which I seriously recommend against using it). So change your wiring to match the software.:
RO -> RX0
DI -> TX1
RE/DE -> D2
2- Should I connect the GND to the FG?
What's an FG? If it's the Omron, the answer is no. RS-485 signals are transferred differentially, so GND is not necessary and will make things only more complicated.
3- Do I need to remove the Arduino from PC's USB after programming? I'm asking it, because I'm using the HardwareSerial.
No, you don't need to but you also cannot use the PC connection (p.e. for debugging the software).
4- In this scenario, the DIP switch setup should be as illustrated in image 03? Only DIP4 is OFF.
With the information I have I'd guess this is correct.
5- Can I test it just forcing A641.00 and checking the DM Area Data, as shown in Image 05
I hope you know the Omron well enough to answer that question. Otherwise ask the vendor. The Arduino should answer if the Omron is asking for the values.