Let's look at it in order.
As a Theurgy Wizard, you reach level 14 and earn the Arcane High Priest, which gives you the level-17 Arcana skill, Arcane Mastery. This allows you to add a level 6, 7, 8 and 9 spell to your domain spell list, so that they are always prepared.
Each domain has a list of spells – its domain spells-
that you gain at the cleric levels noted in the field
the description. Once you've earned a domain spell, you still have
have it prepared, and that does not count against the
number of times that you can prepare each day.
If you have a domain spell that does not appear on the
list of cleric spells, the spell is nevertheless a cleric spell for you.
The section on spellboxes in Chapter 10 states the following:
It does not matter how many spells a launcher knows or
preparehe can only launch a limited number of
spells before resting. Manipulate the fabric of magic
and channeling his energy into a simple spell is
physically and mentally distressing, and higher level spells
are even more. So, every class of caster
the description (except that of the warlock) includes a table
showing the number of spell locations each spell level a
character can use at each level of character. For example,
the third-level wizard, Umara, has four top-level spell locations
and two 2nd level locations.
When a character casts a spell, he spends
a slot of the level of this spell or more, actually "filling"
a slot with the spell. You can think of a spell location as a
groove of a certain size – small for a top-level location, larger
for a higher level spell. A 1st level spell is written in a slot
of any size, but a level 9 spell is only appropriate for a level of 9
slot. So when Umara launches a magic missile, a 1st level
fate, she passes one of her four top-level slots and has
So, you have prepared these spells, but you do not have the spell locations to use them: at the 14th level, you have a slot for the 6th and 7th levels, but not the 8th and 9th. As you said, you have to earn 3 extra levels in a class of spellcasters to get those slots (what about the 17th level wizard or the 3 tier mashup? levels in another).
Assuming you take these three multi-grade levels and then reach level 14 of the Theurgy Wizard, you would have spell locations to cast your prepared spells and thus be able to use them at that level.
The potential argument against this comes from the section on multicast spell locations:
If you have more than one spell class, this table
could give you spell locations of a higher level than
the spells you know or can prepare. You can use those
slots, but only to cast your lower level spells. If a lower-level spell that you cast, such as a burning hand, has a
improved effect when you use a top-level location, you
can use the improved effect even if you do not have
all the spells of this higher level.
However, in this case, you are assigned a number of slot locations for the spells you want. can prepare, because you always have them prepared. The rule that clerical domain spells still appear in your prepared spell list bypasses the rule that you can only choose to prepare a spell if you have a location of the same level or higher. You do not choose to prepare it.