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I just wanted to confirm if I am interpreting the rules correctly. Is the following correct?
Familiars have an "effective" HD equal to either the HD of the base creature, or their master's level. For feats and skills, a familiar does not gain feats or increase its skills as it's "effective HD" increases, since this is not an "actual" change of HD.
Hit Dice: For the purpose of effects related to number of Hit Dice, use the master's character level or the familiar's normal HD total, whichever is higher.
Hit Dice represent a creature's general level of power and skill. As a creature gains levels, it gains additional Hit Dice. Monsters, on the other hand, gain racial Hit Dice, which represent the monster's general prowess and ability. Hit Dice are represented by the number the creature possesses followed by a type of die, such as “3d8.” This value is used to determine a creature's total hit points. In this example, the creature has 3 Hit Dice. When rolling for this creature's hit points, you would roll a d8 three times and add the results together, along with other modifiers.
Next, give the creature additional feats. Creatures gain one feat at 1 Hit Die and one additional feat for every 2 Hit Dice above 1. Most additional feats should be focused on increasing the creature's combat abilities, but metamagic feats and skill feats are also possible choices depending on the creature's role.
It is my not so humble opinion that feats come as a direct result (i.e. as an effect) of hit dice as shown above. As such feats should be gained whenever the effective hit dice are an odd amount. Same with skill points except every level.
However most people will tell you that familiars don't gain increases in skills or feats. Since even though feats and skills are effects of hit dice, the "actual hit dice" don't increase -- even though the familiar rules specifically state that you treat the familiar as if those hit dice had been gained for all effects related to hit dice.
Familiars and animal companions do not gain any skills, feats, or racial bonuses that are not listed on the chart in the character class section or by the specific animal pargraph in the class section of the core book.
The "Pets" are a sub-set of the player's character and not an NCP or PC in their own right. If a DM wants to house rule in allowing pets to be more powerful by giving them extra skills and feats then they should consider also allowing the fighter or barbarian to have extra stuff too.