Can a familiar learn other skills aside from what skills are listed in its entry in the MM's? I can't find anything on this topic.
It'll have human or greater than human intelligence eventually, so there's no reason why it shouldn't be able to learn other skills. Obviously, anything silly, like 'Craft Gemcutting' is probably not going to be useable, but you should be able to get any skill (as a cross-class skill) your AC could reasonably use.
The Creature in question, which starts as an Animal Companion and get's to add in Familiar abilities, is a new type of creature called a 'Companion Familiar'. Every DM I've played under has interpreted that to mean that any feat/PrC/etc that advances or expands on either Animal Companion OR Familiar will work, but I've seen (valid) arguments online for why they shouldn't.
If you CAN get it and your DM let's you treat your CF as a familiar for feats, the Enspell Familiar feat is the best option for abusing Share Spells, as it extends the range out to 1 mile (which is as far as the familiar can go away from you anyway). At your level, with your cohort, you could easily have Bite of the WereTiger persisted to both yourself and your companion, along with all your other buffs, practically all the time.
If you cannot get EF, Companion Spellbond allows some abuse of Share Spells, but the AC is limited to a range of 30ft, IIRC. Spell-Linked Familiar is another option, if your DM let's it count, to allow it to buff itself or give you some action economy with your spells.
Even if you go with something simple like a Hawk that shares your space, you can still really really abuse the buff mechanic, or share spells.
Body of War, for example, is a spell that very much sucks, generally, but Cast it on your companion familiar, and now you've got a potent additional tank in the party.
