It doesn't change into a different creature because of its HD. Its always a small air elemental. It just has a higher effective HD for effects relating to HD.
Edit: It would be awkward to have a huge elemental be a familiar I would think, when every other familiar is smaller than their master and easy to bring. You can always talk to your GM about handling it in your own little way.
and they don't "gain" HD neither : they use their master's level instead of HD (if it's higher) for all effects that are based on the target HD (for example : sleep, Circle of death, ...).
They are not augmented for real however : their HP is always equal to half those of the master, and their BAB, Base bonus to save and ranks in skills are those of the master. A familiar does not gain feats or bonus to attributes (except the new INT).
I'm at work right now (no books!) so I can't be absolutely sure, but I think you can gain some HD from the Boon Companion feat.
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Can they gain skills or feats somehow? I'm looking at stuff from the Animal Archive and it talks about "feats for familiars" and Familiar Archetypes that grant new class skills but don't elaborate any further.
I believe the animal archive also brings up that they can exchange their standard starter feat for one of the familiar feats (I don't have it in front of me, so I might have read it elsewhere). There's also the Beast-Bonded Witch archetype that can give up a feat to give one to their familiar.
I wish they could get more feats in general--my groups usually houserule it to let them get more as they level as if they were getting HD.
So, small air elemental uses the base saves of Fort+2, Ref+2, Will+0
Actually, the +2 fort and ref and +0 will is for standard familiar, as they have a single HD of animal. For improved familiar, base save bonuses are often higher.
For your example : a small elemental uses its base saves bonuses (+3 fort and ref, +0 will) or those of the masters if higher, and add its own attribute bonus to it.
So for a 5th level wizard (to be able to select such a familiar), it will have base save bonuses of +3 FORT and REF (=> those of familiar, as they are better than the +1 of the master), and +4 WILL (=> the one from the wizard, as it is better than the +0 of the familiar).
Adding its attributes, it will have +4 FORT, +6 REF, +4 WILL.
By being a familiar from a 5th level wizard, it will also have 8 INT instead of the standard 4 (and will gain retroactively 4 skill points to add in its own skills ranks), and +3 to its natural armor added to its own (for a total Natural armor bonus of +6), acquire several familiar power (improved evasion, communication with master, ...), and its whirlwind will raise from DC 12 to DC 13 (because it is an effect based on HD, and thus now based on wizard level instead).