Correct. Other than spell casting being useful for buffing the eidolon, and cha setting the number of summoning SLAs per day (okaybthe second doesn't involve the eidolon, but seems worth noting).
Correct, except that charisma also affects your Summon Monster SLA.
Many of the spells that the summoner casts are intended to benefit the eidolon in some way, so in that sense increasing the summoner's spells per day via a higher charisma is good for the eidolon.
Totally agree that having more CHA is best for the summoner, but I've just been thinking I'm misreading the Eidolon because I keep thinking that the summoner's attributes affect it somehow but I couldn't find them. Thanks for the double check.
You may have unwittingly been thinking about Synthesist.
The only things passively affecting your eidolon are your magic items (for slots where you both were something).