Yes, you increase Animal Companion's ability score when advancing it by HD.

The rules for Animal Companion bonus HD state explicitly which benefits it receives as normal for advancing a monster's Hit Dice. That allows to conclude, following the logic of the D&D system, that other benefits are not gained as normal, and therefore size and ability scores are not increased per bonus Animal Companion HD.

However, this article explicitly says that Animal Companions advance as normal per HD, except size. This raises a question of source precedence, and as for me, the mentioned WotC article specifically regarding Animal Companions is a primary source.

A quote from the article in question, for convenience:

Bonus Hit Dice.

The animal companion gains extra Hit Dice as noted in the table on page 36 of the Player's Handbook. The companion gains the full benefit from increased Hit Dice, including increased base attack, base saves, skill points, and feats. The animal companion, however, does not increase in size (any more than you do when adding Hit Dice for your class levels).

Note: Pathfinder Druid has this covered.

Your animal companion, like any other creature, can attempt to use any skill that does not specifically forbid its use without training, and does so just as any other creature does. However, its ability to do so may be restricted by its lack of training (such as in the case of Knowledge skills), and/or its lack of ability to communicate (such as in the case of Diplomacy). As well, there may be circumstance bonuses or penalties (there's a section on those in the core books) that apply, and I'd consider it highly likely that your DM may choose to make house rules about skills such as Craft, Spellcraft, etc.

For all future reference, the above considerations apply to anything with hit dice - they can try to use any skill that they're not specifically forbidden from using by either training or a creature trait (such as Mindless).