[RPG] What checks can familiars Help with

RAW (Rules As Written), it looks like it according to the rules for familiars and looking at improved familiar, improved familiars do not give you a bonus to skills. Not that this is a terrible thing; some of those improved familiars have super nifty abilities! They just don't give you more skill points to a skill. (Some of them could just perform the skill for you!)

However, one could argue that the improved familiars should grant skill bonuses, as per the "approximating familiars" section in the first link. This is all subject to DM approval; I doubt many of them would allow it.

Spells with a range of self cannot be cast via familiar

The restriction you quote acts exactly as it says: you can only cast touch-ranged spells through your familiar. "Self" != "touch," so you're out of luck.

As for touch spells that damage someone, you correctly got forbiddance, inflict wounds, and shocking grasp, but you missed symbol and glyph of warding. But they're strange edge-cases—the damage isn't done when the spell's cast, but rather when the glyph is triggered. There's also plane shift which, when cast on an unwilling target, is a spell attack which doesn't do any damage.

Magic stone—from the Elemental Evil Player's Companion—is another edge-case: you cast the spell on the stones via touch, but then make a ranged spell attack with them to damage someone.