A vestige of ASTAROTH, DIABOLUS OF BAATOR, when bound by a Binder grants, among other things an ability
Your spells, spell-like abilities, and magic item abilities with the fire descriptor deal half fire damage and half vile damage. You can cast fireball, as per the spell, three times per day (caster level equals your effective binder level).
An Anima Mage has the ability of
Exploit Vestige (Su): At 2nd level, you learn how to use a bound vestige to boost your arcane spellcasting. With this ability, you can choose to forego gaining one of the vestige's granted abilities in order to gain one additional arcane spell slot per day of any level up to the highest you can cast. Thus, if you bind the vestige Dantalion (see page 28), you could give up the read thoughts ability he grants you in exchange for a bonus spell slot that day.
1.
Assuming a character has entered the Anima Mage via the Precocious Apprentice trick, and now is Wiz 1/Binder 3/Anima Mage 2, with Improved Binder trick he can Bind Astaroth.
Can he gain 3rd level spell slots via Exploit Vestige?
On one hand he "can cast fireball, as per the spell"
On the other hand, that is not actually a spell, it's a Su ability that grants him the ability to cast fireballs. It's also the question of timing -- does he have the ability to cast fireballs when choosing to use Exploit Vestige or do those things happen somehow concurrently?

2.
Let's say it's instead a somewhat cheesed abomination of Warlock 2 (Precocious Apprentice)/Binder 1 (Improved Binding)/Anima Mage 4
They summon Astaroth, surrendering one of his abilities to gain a spell slot.
Now they have either a 2nd level slot and 3rd level spell slot, or two 2nd level slots, and a single spell they know. What are some cool ideas for the uses of those spell-slots?
Could they use a runestaff? Get new Spells Known somehow?