Unfortunately, it might be a hardware problem, such as graphics hardware, power supply, logic board...
Here are some things you can try, in case the cause is less serious.
Reset PRAM, if you have not already
http://support.apple.com/kb/HT1379
Also reset SMU (power management). This is the procedure for iMac G5.
http://support.apple.com/kb/HT1767
When you connect it back to power, I saw in the video that you were using a direct connection to a wall outlet, by itself. That's good, to rule out a faulty (or over-crowded) power strip. Be sure to disconnect ALL peripheral devices; start it up with only the power cord connected, then add just a standard keyboard and mouse. If that makes a difference (and it starts up properly), run it that way for a while to see if it continues.
Check the vents along the bottom (and elsewhere) for accumulation of dust, blocking airflow. That could cause overheating, which may result in video issues. If this is the older type of iMac G5 that is easy to open up, you can also check it internally for dust.
If the PRAM battery has never been changed, it could be weak or dead. That can cause odd startup issues, but usually not like the issues you have. The PRAM battery for an iMac G5 looks like this.
http://eshop.macsales.com/item/NewerTech/CR2032/
The fact that it works better in Safe Mode makes me think it is related to the graphics hardware. In Safe Mode, the graphics hardware is not being used to full capacity, so it might work more reliably (if it's becoming faulty). For similar reasons, it could be the power supply, which may be less "stressed" in Safe Mode.