Can I power on my old MacBook with a no-lights battery? I've let my MacBook's battery go far too long past its life because it lives at a desk and never moves. It's a MacBook Pro 4,1. 
On MacBook Pro 3, i was always able to boot without battery, but this one will not boot after SMC reset with or without battery installed. 

Pressing the battery's charge checker button does not light up any LEDs now. Previously it would rapidly blink the first LED , indicating it needs replacement. 

Is there any way to boot? (I'm aware it will run at lower clock speed)
Normally you should be able to, and even Apple says so, by providing a warning and a CPU clock speed reduction.

If it does not work on your computer.....

You might be having a problem with the MagSafe adapter or SMC. 

Your machine should turn on normally whether or not the battery is in it. 

The only problem you'll have with the battery removed is that machine will run slower - but that's by design. 

When you get the new battery, make sure you try another MagSafe adapter to be safe.

Try another SMC reset by Holding the power button for 5 or 10 seconds.