I'm beginning to think that. I created a bootable disk on a flash drive, but when I try to install it the only choice for where to install is the flash drive itself, not the hard drive. See attached.
Click back and try opening the Disk Utility on the top bar menu.
Look for the HDD and try partitioning it. (Create a volume)
If the HDD is still good, it might have been wiped out even from partition level. Therefore the iMac can't recognize it for OS installation.
You would need to partition it (create a volume), then format the HDD to proper format (APFS, perhaps) before High Sierra installer can recognize it as a volume to install OS onto.
If you can't see the HDD from Disk Utility, then congratulations, you now have the chance to upgrade to an SSD against your will.
To get the iMac recognize Bluetooth devices like keyboards and mouses, they must be paired with it for the first time. And this can only be done after entering Mac desktop.