The Aether Puppet's text is a reference to the object previously mentioned in the description and the additional properties it gets from being a construct of said size:
Creating an animated object of a different size than Medium can be done simply by adjusting the object’s size (and thus adjusting its Strength, Dexterity, natural armor bonus, and size modifier to attack and AC as detailed on page 296) and Hit Dice.
This means the following:
The normal object is a Medium construct (or smaller) with all its statistics listed here.
If it is made of metal or stone, it will automatically get the ability
- Metal (Ex, 2 CP): The object is made of common metal. Its hardness increases to 10, and it gains a +2 increase to its natural armor bonus. [no Mithral or Adamantine yet]
- Stone (Ex, 1 CP): The object is made of stone or crystal. Its hardness increases to 8 and it gains a +1 increase to its natural armor bonus.
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At 12th level, that object inherits all of the above and gets the following additional changes:
- size: Large (or smaller)
- hitpoints: 52 (4d10+30) [Medium only has 36 = 3d10+20]
- Slam attack: +5 (1d6+3) -> +9 (1d8+7) [x2, see Additional Attack]
- Additional Attack (Ex, 1 CP): Gains an additional slam attack.
- apply the Creature Size modifiers: -1 size modifier on attack rolls & AC, +1 special size modifier on CMB & CMD, +8 STR, -2 DEX, +2 natural armor [ignore CON, as constructs don't have that]
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At 14th level, that object inherits all of the above and gets the following additional changes:
- size: Huge (or smaller)
- hitpoints: 78 (7d10+40)
- Slam attack: +9 (1d8+7) -> +15 (2d6+11) [x2]
- apply the Creature Size modifiers [in brackets: compared to Medium]: -1 [-2] size modifier on attack rolls & AC, +1 [+2] special size modifier on CMB & CMD, +8 [+16] STR, -2 [-4] DEX, +3 [+5] natural armor
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At 16th level, that object inherits all of the above and gets the following additional changes:
- size: Gargantuan (or smaller)
- hitpoints: 115 (10d10+60)
- Slam attack: +15 (2d6+11) -> +20 (2d8+15) [x2]
- Metal (Mithral) (Ex, 4 CP): gain hardness 15 plus a +4 increase to natural armor. [only if object is made from mithral]
- apply the Creature Size modifiers [in brackets: compared to Medium]: -2 [-4] size modifier on attack rolls & AC, +2 [+4] special size modifier on CMB & CMD, +8 [+24] STR, +0 [-4] DEX, +4 [+9] natural armor
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At 18th level, that object inherits all of the above and gets the following additional changes:
- size: Gargantuan (or smaller)
- hitpoints: 151 (13d10+80)
- Slam attack: +20 (2d8+15) -> +23 (4d6+19) [x2]
- Metal (Adamantine) (Ex, 6 CP): gain hardness 20, and receive a +6 increase to natural armor. [only if object is made from adamantine]
- apply the Creature Size modifiers [in brackets: compared to Medium]: -4 [-8] size modifier on attack rolls & AC, +4 [+8] special size modifier on CMB & CMD, +8 [+32] STR, +0 [-4] DEX, +5 [+14] natural armor
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The abilities Metal (Mithral/Adamantine) and Additional Attack are gained regardless of size at the respective levels, but the other changes depend on the creature's size.
The attack roll modifiers are included in the calculation.
The Fly & Stealth skill would also be modified accordingly.
Reach depends on the shape of the animated object (tall/long), so you'd have to look that up in this table.
I suspect you are reading the rules text a little wrong, as you consider it one block of text instead of the different options provided. In order to remedy that, I will add a little text formatting to the rules text of Aether Puppet and [quick notes in brackets] to make it more easily understandable:
[Fluff text]: You use strands of aether to animate an object.[Base Effect A]: You can animate a Medium or smaller object as an animated object (Pathfinder RPG Bestiary 14) with no Construction Point abilities (except for stone or metal [common], if the object is constructed of one of those materials).
[Base Effect B]:Each round on your turn, you must take a move action to guide the object, or it reverts to its inanimate state. By accepting 1 point of burn, you can pour a bit of your own sentience into the aether puppet, allowing the effect to persist for 1 round per kineticist level without requiring additional actions.[Option 1]: At 12th level, you can animate a Large or smaller object, which gains all the abilities of the Medium animated object plus the additional attack ability.
[Option 2]: At 14th level, you can animate a Huge or smaller object, which gains all the abilities of the Large animated object.
[Option 3]: At 16th level, you can animate a Gargantuan or smaller object, which gains all the abilities of the Huge animated object, as well as the metal (mithral) ability if the object is made of mithral.
[Option 4]: At 18th level, you can animate a Colossal or smaller object, which gains all the abilities of the Gargantuan animated object, plus the metal (adamantine) ability if the object is made of adamantine.
The [Base Effect A] part of the text describes the effects of this Wild Talent, i.e. that you can animate a Medium or smaller object as an Animated Object according to the Bestiary; it also says how to deal with the construction points of such an Animated Object.
The [Base Effect B] part tells you the duration of the animation, requiring a Move Action every round to sustain it; this part of the text also includes an option on how to accept Burn to give it a duration without Move Actions.
The next parts are essential: The Aether Puppet wild talent offers different options depending on the Kineticist's level, i.e. you have to choose one of these options. Each of these options replace the [Base Effect A] part, the Medium object.
For example, when you choose [Option 3], you only get the [Base Effect B] ("duration") with the alterations of [Option 3], but you do not get anything from either [Base Effect A] (since it's replaced), nor [Option 1] nor [Option 2] unless it specifically says so.
This is why the text includes "... which gains all the abilities of the Medium/Large/Huge/Gargantuan animated object."
If that part of the text were omitted, then the Large animated object would not get the Stone/Metal ability of the Medium object;
the Huge animated object would neither get the Stone/Metal ability of the Medium object, nor the Additional Attack ability of the Large animated object; and so on.
The text merely tells you that each new option integrates all the abilities gained by the previous options.
I still do not understand why at level 14, for example, you write
Slam attack: +9 (1d8+7) -> +15 (2d6+11) [x2]
At level 14, the Huge animated object gains the abilities of the Large animated object (Additional Attack + all Medium animated object abilities).
Therefore the Slam attack goes from "+9 (1d8+7)" to "+15 (2d6+11)", after applying all the changes from the Large to the Huge animated object:1) Construct type creatures have a full-BAB progression, i.e. BAB = HD; therefore the 3 additional HD increase BAB by +3.
2) The additional +8 in size modification increase the attack roll bonus by another +4.
3) The additional -1 size modifier penalty reduces the attack roll by -1.
=> +9 +3 +4 -1 = +15
The damage dice size increases by one step, from "1d8" to "2d6" and the additional +8 Strength add another +4 dmg to the Slam attack.
=> 1d8 -> 2d6 ; +7 +4 = +11
The [x2] is from the Additional Attack ability of the Large animated object.
=> "Slam attack: +9 (1d8+7) [x2]" changes to "Slam attack: +15 (2d6+11) [x2]".