iMac G5 won't startup

Sounds to me like the HDD is failing or failed. The clue is that it failed to mount via TDM. If you have another disk drive to use, then see if it will boot from it. Or just replace the existing one with one known to be working.

How did it tell you the machine cannot boot from a USB drive unless your machine does not have USB 2.0 ports. If it does then it should boot from USB. You must use Option booting, however.

I doubt it since you could boot some older Macs from USB, but not others. Perhaps your's does not allow booting from USB drives. Try using Firewire or connecting the drive internally on another machine to test if the drive really is defective.

What version of OS X are you installing on the computer? How was the other disk you tried formatted? You can't boot the computer from a PC disk until you have re-partitioned it GUID and formatted it Mac OS Extended, Journaled. If the PC disk was formatted NTFS, then you need to reformat it using FAT32 or MSDOS on the PC from which it was removed or you can try to re-partition it on the Mac using MBR and formatting it MSDOS. This latter may not work since OS X cannot write to an NTFS disk.

When you partitioned the internal drive before installing did you set it to use APM or GUID? You should use the former. Although you can install on the GUID partitioned disk you cannot boot from it. Booting in Safe Mode may work, but if you cannot boot normally then there is a problem either with the disk or the computer. If you were able to boot from the installer DVD, then I'm inclined to go with the disk.

'Perhaps you should. Do this:

Perform an Erase and Install

1. Insert the OS X DVD into the optical drive and restart the computer.
2. Press and hold down the "C" key after you hear the startup chime. Release the key when the progress indicator appears below the dark gray Apple logo.
3. Select your language after the installer begins and click on the Continue button.
4. After the menu bar appears select Disk Utility from the Utilities menu.
5. After Disk Utility loads select your hard drive (this is the out-dented entry with the mfgr.'s ID and size) from the device list. Note the SMART status of the drive in Disk Utility's status area. If it does not say "Verified" then the drive is failing or has failed and will need replacing. SMART info will not be reported on external drives. Otherwise, click on the Partition tab in Disk Utility's main window.
6. Click on the Options button, set the partition scheme to APM, then click on the OK button. Set the number of partitions from the Partition Scheme's dropdown menu to one (1). Set the format type to Mac OS Extended (Journaled.) Click on the Partition button. Wait until the volume mounts on the Desktop.

Did you insert Disc 2? If so what happened? If still not booting, then you need to start again. See the following:

Snow Leopard Erase and Install

Be sure you back up your files.

1. Insert Snow Leopard DVD into the optical drive and restart the computer.
2. Immediately upon hearing the chime hold down the C key.
3. Release the key when the Apple logo appears and wait for the loading to finish.
4. After the installer, loads select your language and click on the Continue button. When the menu bar appears select Disk Utility from the Utilities menu. After Disk Utility loads select the out-dented disk entry from the side list (mfgr.'s ID and drive size.) Click on the Partition tab in the Disk Utility’s right window. Set the number of partitions to (1) from the dropdown menu. Click on the Options button and select GUID, click on OK, set the format type to MacOS Extended, Journaled. Finally, click on the Apply button.
5. After formatting has finished quit Disk Utility. Continue with the OS X installation and follow the directions.
6. When the installation has finished the computer will restart into the Setup Assistant. After you complete the Setup Assistant you will be running a fresh installation of Snow Leopard. Open Software Update and install the recommended updates.
7. If you are planning to sell or give your computer away, then do the following: After you reformat your hard drive and reinstall Snow Leopard, the computer restarts to a Welcome screen. To leave the Mac in an OOTB (Out Of The Box) state, don't continue with the setup. Instead, press Command and Q keys to shut down the computer. The Setup Assistant will guide the new owner through the setup process after he/she turns on the computer.
8. Download and install Mac OS X 10.6.8 Update Combo v1.1. if needed. It is not needed if you are selling or giving away the computer.

I have a mid 2008 Mac Pro (big silver desktop) running Snow Leopard - and have had the slow startup off and on for awhile (november 2015) since I deleted some old files and had a major freeze. After a few months -- found a post on this site about a startup taking hours not minutes - so decided to be patient and assume its doing something - sure enough - It made it through the grey and blue screens -- to desktop. If the desktop does not load in the correct sequence- I restart until it does. If I have an application that requires force quit or crashes - I immediately relaunch my finder.

One issue is the Spotlight indexing - so I moved all my drives into private - as I don't use spotlight. If you keep resetting the pram - your preferences may lose your selected startup drive - so use Option after the chime to select the startup drive -- that helped me until I stopped wasting my time resetting the Pram.