Can Familiars Get Feats?

That the familiar cannot gain feats, so long as it is a familiar.

Do familiars gain hit dice?

In short: Familiars gain hit dice. Number of hit dice is equal to the caster's appropriate class level (or the familiar's normal HD total, whichever is higher).

Do familiars gain levels?

Familiars do not gain class levels, nor do they advance via the Monster Advancement rules from the Bestiary. Instead, Familiars advance with the level and/or class level of it's master, and follow their own rules for that: Hit Dice: [... M]aster's character level or the familiar's normal HD total, whichever is higher.