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Do I have this correct?:
Fodder grade automaton, 10XP * 10 = 100GP base cost.
Tiny size gives 1/16 base cost, 6.25GP.
Add a 4GP power stone, weighing 2 pounds, to power the device 4 days between charges.
So 10.25 GP total.
When using the device under Artifice school power, Activate Automaton, use the device's fodder stats for attack and damage?
For Guardian Automaton power, use my character's stats for damage?

You have the costs for your automaton totally correct. Nice work.

You use your stats for both powers, since they are your powers. When one of your powers refers to a stat, unless otherwise noted, it means you.

A minor note: a tiny creature has 1/4 normal carrying capacity (pg 19, weight allowance) and the automaton has a Str of 2 (since it is a fodder, all abilities are 2). So its maximum carrying capacity is 5lbs (2 Str x 10 x 1/4) and it should have no problem carrying the power gem, but it is carrying a lot of weight, so will have trouble carrying much else.

Additionally, if you want to command the automaton to do things other than your Artifice powers or act using its own powers, you'd have to spend CP on the Companion - Automaton talent.

Carrying capacity and limited use from not being a campanion aren't a problem in this case, the intended use is the automaton acting like a Dancing Weapon, mixed in with a couple proper Dancing Weapons.

Dancing Chainsaw?

Not quite the image I had in mind. Think whirling short swords darting across the battle field. But if it fits the character I can see dancing chainsaws working.