That's quite a lot of questions! Honestly though, I think your biggest issue is that you have not defined a threat model. You need to understand your adversary, their capabilities, the value of your assets, etc. It's far more likely that someone will simply plug in a hardware keylogger than it is that they will physically open your computer and replace your BIOS. Attackers will always choose the simpler attacks over the more sophisticated ones, and your adversaries are no exception.
Encryption cannot protect against an attacker who has unrestricted physical access to the device doing the encryption. This is not a limitation of encryption itself, but a limitation with how it can be used. VeraCrypt and similar encryption utilities can only provide data-at-rest security which means that, when the system is powered off and all an attacker has is the raw encrypted data, they will not be able to decrypt it without the key. Make sure your system is attended and physically secure.
I don't want the VeraCrypt hidden volume I created to be damaged. If I mount the hidden volume and transfer files there'd no damage, and if I enable the option 'Protect hidden volume against damage caused by writing to outer volume ' In the Mount Options dialog window, and write to the outer volume, there is no damage to the hidden volume, is that correct?
That is correct. If you enable "protect hidden volume", then you can write to the outer volume safely. If you are forced to give up your key under duress, you would only provide the outer volume key and not both, which would cause any writes to corrupt the hidden volume. This is unavoidable.
if someone has access to a Veracrypt encrypted flash drive, they wouldn't be able to mount or write information to it, and BadUSB, flashing malware into the firmware wouldn't be possible?
Encryption does not block writing, but it does mean that the data that is written will appear as gibberish when you decrypt it. As for flashing firmware, encryption does not prevent that. Firmware is kept separately on the flash drive and is not and cannot be encrypted by VeraCrypt.
The only possibility would be to physically open up the flash drive and insert something into it? How likely is that to happen by someone who isn't the government and how much it would cost?
It would be far easier to modify the firmware, but if the drive does correctly disable writing to firmware, then it would be possible to physically modify the firmware chip. No one can tell you the likelihood of this happening, but the cost would be extremely low. Note that modifying the flash drive alone is incapable of circumventing encryption.
I assume that it would be different for a PC with an encrypted hard drive as they would be able to flash malware into the firmware if the BIOS isn't password protected, but what if the BIOS is password protected but you can still use the multi device boot option and boot a Linux operating system from a flash drive? (I know that PC does that). Does that mean that they could flash something into the firmware? How do I test if it's possible on that PC?
The BIOS password does not protect against firmware modification. See this answer.
If four isn't possible, the only possibility would be again to open up the PC and insert something into the hardware, is that correct? How likely is that to happen by someone who isn't the government and how much would it cost?
It would be easier to modify a PC because it is larger and has more components. It would be very cheap, but again, no one can tell you how likely that scenario is for your particular threat model.
Is there really no way to check if they inserted something into the firmware or to check if they inserted something into the hardware? How do you keep your information safe if your PC or flash drive could be compromised with no way to check if it has been compromised, aside from getting a new machine?
It is possible to verify firmware, but it can be complicated to set up. I explained how this can be done on an answer to another question. At the bottom of the answer are a list of further resources.