How do I install Linux on a 2007 iMac that has its operating system uninstalled?
I haven't used this process to install Linux Mint, but I have used it to install other operating systems on earlier iMacs and MacBook Pros. So I offer this answer in the hope it works for you, but if not, it may help others.

I see two ways to install an OS from a bootable USB on earlier Macs that didn't support booting from USB:


Start the destination Mac in Target Disk Mode
Remove the HD from the destination Mac and insert it into another Mac or external case


Obviously the above options aren't for everyone. Both options requires access to another Mac that will boot from USB and the second one requires the user being comfortable with the process (in some cases the process would be a nightmare, in others fairly straightforward).

Target Disk Mode

The following is a general guide on what's involved:


Start the 2007 iMac in Target Disk Mode by holding down the T key during startup. You'll know when it's finished by seeing an icon floating around the screen. 
Use an appropriate FireWire cable to connect the iMac to another Mac capable of booting from USB. (Note: You may need to use an adaptor if the other Mac does not have native FireWire support - e.g. Thunderbolt > FireWire adapter).
Insert the bootable USB (e.g. for Linux Mint) into the second Mac (i.e. the one capable of booting from the USB).
Start the second Mac and immediately hold down the Option key until you see the startup disk manager.
Select the bootable USB from the list and press Enter.
Follow the steps to install the OS (e.g. Linux Mint) onto the hard drive of the first Mac in Target Disk Mode (e.g. the 2007 iMac).
When complete, shutdown both computers and disconnect the FireWire cable between the two Macs.
Start the first Mac normally (hopefully to see it boot from the OS you just installed).


Remove the HD

This involves removing the hard drive from the first Mac and inserting it into another Mac or an external case. However, choosing this option may depend on the model of your first Mac (e.g. removing a hard drive from a Mac Pro is a lot easier than removing it from an iMac) and may also depend on the model of your other Mac (e.g. a HD from an iMac will not fit into a MacBook).

So, this option will be easier if you use an appropriate external case, and if the original Mac is not hard to work with. If you use an external case, then the process is similar to using Target Disk Mode, because a Mac started in Target Disk Mode is effectively an external hard drive.

Obviously I can't cover all possible scenarios here, but this option is not so easy for an iMac (not even the 2007 model used by the OP). But for users comfortable with doing this, the process is broadly as follows:


Remove the HD from the first Mac
Insert the HD into an appropriate external case
Connect the external case to the Mac capable of booting from USB
Insert the bootable USB (e.g. for Linux Mint) into the second Mac (i.e. the one capable of booting from USB)
Start the second Mac and immediately hold down the Option key until you see the startup disk manager.
Select the bootable USB from the list and press Enter.
Follow the steps to install the OS (e.g. Linux Mint) onto the correct hard drive (i.e. the one from the other Mac)
When complete, shutdown your Mac 
Disconnect the external case
Remove the HD from it and insert it back into the first Mac
Start the first Mac normally (hopefully to see it boot from the OS you just installed).


Note: If you're in a situation where you can install the hard drive into another Mac, then at Step 2 you would install it into the Mac instead of the external case and you'd also skip Steps 3 and 9.