You seem to be misunderstanding the function of resistors. You must have heard this: “A resistor turns a voltage into a current, or a current into a voltage”?
A potential divider can indeed ‘program’ a specific voltage at it’s junction. But any load at all (even a small one), will change that voltage because it changes the current.
Let’s say you use a 100R and a 500R divider, with the bottom of the 500R at ground and the top of the 100R at 6v and the junction feeding the Picaxe. (aside from the fact that this is a mad waste of power), You would be right in expecting there to be exactly 5v at the junction. But as soon as you turn on the Picaxe, current flows through it. It’s almost exactly like putting another resistor in parallel with the 500R. So your divider voltage changes (drops).
I say ‘almost’ exactly, because unlike a resistor, the current used by the Picaxe will change from moment to moment depending on what it’s doing at the time. EG: You light a LED. That’s an additional 10mA etc. So the divider voltage is going to be changing unpredictably.
Variations in the Picaxe supply voltage will cause problems. At best, you will not be able to use the ADC inputs very well. At worst, the Picaxe will crash.
If you eliminate the 500R and just rely on the Picaxe itself as the bottom leg of the divider (I think this is what you were suggesting), thing’s will be even worse.
The voltage across a resistor will change according to how much current is flowing through it. Easy to calculate using Ohms law. V=IR, when V is the voltage across the resistor (in Volts), I is the current flowing through it (in Amps) and R is the resistor value (in Ohms).
A diode behaves very differently. It does not follow Ohms law. When forward biased, the voltage across it remains fairly constant. So variations in the load (Picaxe current) will not change the voltage across the diode appreciably.
Your question “How can I find the resistance the PICAXE chip” is a reasonable one. But would be better phrased “How can I determine what current will be flowing through the Picaxe chip?”. And of course the answer is, you can’t really. It changes from moment to moment. You can of course measure the current and calculate the equivalent resistance, but what would be the point when a millisecond later it will be different and you would have to calculate it again. This is essentially what Voltage Regulators do, they constantly adjust the current to maintain a constant voltage for the Picaxe.