#1) Playstation1(One)
PS1 uses a PSX to RCA cord to connect to the Playstation to most TV channels. Does your cable look like this one -->
http://www.amazon.com/Playstation-PS2-PSX-RCA-Cable-2/dp/B000GACEHQ ? If so, you can buy RCA extension cables like this -->
http://www.amazon.com/LK-131035-Male-Female-Extension-Cable/dp/B003V7B6FC/ref=sr_1_6?ie=UTF8&qid=1324318012&sr=8-6#2) Nintendo Game Cube
Same question.
#3) Atari Flashback 3
I don't know about this one. I have never owned one, but if its the same connector at the end (Red/Yellow/White), it should work with the extension cables.
Now, would it be better to buy extension cables for each set, or would it be better to buy whats called an AV switch? An AV switch would have one set of cables that run from it to your tv. All your other devices would plug into the AV switch. The AV switch has push buttons and it redirects the connections from the device plugged into the respective port to the TV.
Say you have your PS1 plugged into port 1 on the AV switch, the GameCube on port 2, and the Atari on port 3. If you wanted to play the PS1, you would push the button that says 1, or if you wanted to play GameCube, push 2, etc. On our primary TV, we are limited to how many RCA/Composite connections we can have so we purchased an AV switch to plug all the devices to it.
Hope it helps!
Jerry