It's usually best to use the most recent printing of anything, and that means using the Spell Compendium version the 7th-level Sor/Wiz spell animate breath [trans] (SpC 11):
For this spell to function, you must have a breath weapon, either as a supernatural ability or as the result of casting a spell such as dragon breath (page 73). When you successfully cast this spell, you imbue the energy of your breath weapon with coherence, mobility, and a semblance of life [for 1 round/level].
Emphasis mine, and I'll address that shortly. One should also use the Races of the Dragon version of the feat Draconic Breath (102):
As a standard action, you can convert an arcane spell slot into a breath weapon. The breath weapon is a 30-foot cone (cold or fire) or a 60-foot line (acid or electricity) that deals 2d6 points of damage per level of the spell slot you expended to create the effect. Any creature in the area can make a Reflex save (DC 10 + level of the spell used + your Cha modifier) for half damage. This is a supernatural ability.
Emphasis mine, and there are two ways of reading that emphasized text.
The creature takes a standard action to convert an arcane spell slot into a breath weapon and uses that breath weapon as part of the same standard action used to convert the slot.
This reading means the breath weapon the feat grants can't be targeted by the spell animate breath. The spell slot is converted into a breath weapon that's instantly used, and, prior to that conversion (if the creature has no other breath weapon) the creature has no breath weapon to target with the spell animate breath. I assume this is the feat's typical reading.
Hence a caster could no more cast the spell animate breath on this reading of the breath weapon granted by the feat Draconic Breath than he could cast animate breath on the breath weapon generated by the 4th-level Sor/Wiz spell firestride exhalation [conj/evoc] (Dragon Magic 67).
The creature takes a standard action to convert an arcane spell slot into a breath weapon, and, afterward, the creature can take a standard action to employ the breath weapon.
This reading is extremely legalistic, and it's unlikely this was feat's intent, but this reading does allow the spell animate breath to be cast on the feat Draconic Breath's breath weapon (even if the breath weapon is made available through expending a 0-level spell). This reading also removes the utility value (such as it is) of being able to spend spell slots to make breath weapons that hurt foes right now. (The creature is essentially forced to take two rounds to use its breath weapon. And, no, that's not a delay expressed in rounds.)
Under such a reading the DM must determine if the creature possessing the feat can either take only one standard action to convert one spell slot into one breath weapon or take several standard actions to convert several spell slots into several distinct breath weapons. Essentially, the DM must determine if the feat is a switch or a dial. (This DM recommends a dial.)
The spell animate breath fails if the caster lacks a breath weapon
Although the spell's duration is 1 round/level, the spell animate breath nonetheless requires the caster to cast it on the caster's breath weapon. Otherwise, the spell fails. The typical reading of the Draconic Breath feat has the feat supplying a momentary breath weapon that's used in the same round upon exchanging the spell slot for the breath weapon, leaving no time for the spell animate breath to be cast.
A spell like the 4th-level Sor/Wiz spell dragon breath [evoc] (SpC 73), with its 1 round/level duration, is a valid target of the spell animate breath, the spell dragon breath even stating that, after the spell's cast, the caster must take a standard action to use the breath weapon and then must wait 1d4 rounds before using the breath weapon again. Until the duration of the spell dragon breath expires, the caster has a for-reals honest-to-Pelor breath weapon, not a fleeting one like that afforded by the typical reading of the feat Draconic Breath.
Question: Does the prestige class dragonheart mage (Races of the Dragon 88-91) change the statistics of the variant Huge fire elemental created by the spell animate breath?
Answer: No. The spell animate breath determines the variant Huge fire elemental's statistics, not the breath weapon employed to create it except insofar as dictated by the spell. Note: The prestige class dragonheart mage is, frankly, less than stellar, so allowing minor changes to the variant Huge fire elemental as a house rule shouldn't upset game balance significantly.
Question: Can the variant Huge fire elemental created by the spell animate breath be created anywhere within what would otherwise be the breath weapon's typical area or must the creature be created somewhere else?
Answer: Absent a range, the most conservative house rule for determining where the spell's effect occurs is adjacent to the caster's space in an area sufficient to accommodate the newly created creature. Note: Using such a house rule, a caster that lacks sufficient space adjacent to him will see the spell's effect fail. The created creature won't, for example, break down walls purely as a result of its creation.
Note: A Huge elemental can be summoned using the 7th-level Sor/Wiz spell summon monster VII [conj] (PH 287). While the spell animate breath has only somatic components and a casting time of 1 standard action, summon monster spells—because of their versatility and splatbook support—are usually far better choices for a sorcerer.
No, you can't cast it using a spell slot.
Spells you can cast because of your race aren't spells you know, and thus can't be spells you prepare. Per the PHB (p. 201, "Known and Prepared Spells"):
Before a spellcaster can use a spell, he or she must have the spell firmly fixed in mind ...
... and it goes on to describe some of the class-specific variations of knowing and preparing spells. But in any case, having a feature from your race or class that allows you to cast a spell isn't the same as knowing the spell.
This is spelled out clearly in the multiclassing rules for Pact Magic quoted in the question (PHB p. 164; emphasis mine).
If you have both the Spellcasting class feature and the Pact Magic class feature from the warlock class, you can use the spell slots you gain from the Pact Magic feature to cast spells you know or have prepared from classes with the Spellcasting class feature, and you can use the spell slots you gain from the Spellcasting class feature to cast warlock spells you know.
Likewise, the Drow Magic trait clearly distinguishes the known cantrip from the merely castable other spells (PHB, p. 24; bold for emphasis mine):
You know the dancing lights cantrip. When you reach 3rd level, you can cast the faerie fire spell once per day. When you reach 5th level, you can also cast the darkness spell once per day. Charisma is your spellcasting ability for these spells.
If the intent of the rules was that Drow also knew the non-cantrip spells, it would say so.
Best Answer
With an ultimate magus, it’s more important to keep yourself as close as possible to a single-classed spellcaster on one side than it is to keep the two even.
Thus you want to circumvent as much as possible ultimate magus’s attempts to progress your lower-level class.
Because at 1st, 4th, and 7th level, ultimate magus advances the class with lower caster level (your choice in the case of a tie), this has a tendency to even out your two classes. If you avoid that, and continue to progress in the more advanced class, you will have more of your highest-level spell slots, which are also higher level than they would be.
Practiced Spellcaster is the solution to taking manual control over which class ultimate magus progresses. Practiced Spellcaster increases a class’s caster level (but no other facet of its spellcasting) by up to 4 (capped at the spellcaster’s HD, so the feat is only beneficial to multiclassed spellcasters). Normally, caster level is nice but not as crucial as things like spell slots or higher spell levels. Ultimate magus switches this up, though.
If a Wizard 4/Sorcerer 1 takes Practiced Spellcaster for sorcerer, his weaker class, his sorcerer caster level is 5th. This puts it higher than his wizard spellcasting’s 4th caster level, so first level of ultimate magus advances wizard. This is despite the fact that the ultimate magus’s wizard spellcasting is, for most purposes, more advanced than his sorcerer spellcasting.
Now you have some control over your advancement, and that allows you to maximize your spell slots.
At 4th level, both classes are even, so you should choose wizard. At 7th, barring finding some other bonus to sorcerer caster level, though, you have to accept some sorcerer progression, which is a shame but not the end of the world. You’ll end up “missing” a total of two wizard spellcasting levels (compared to a single-classed wizard), but gain a total of nine sorcerer levels, plus of course all the rest of the ultimate magus class features.
Otherwise, you need to improve the abilities that grant you bonus spells. Consider the beguiler (Player's Handbook II) instead of sorcerer: you'll use Intelligence for both classes.
Versatile Spellcaster is another feat that will make your slots more flexible.
And pearls of power and rings of wizardry can help by letting you reuse slots. Memento magicka are like pearls of power for spontaneous spellcasters but they are 50% more expensive, which is badly overpriced.