What happens in general varies a lot based on the relative timing of the two effects, the specific effects involved, and the nature of their sources' descriptions. The general answer is that either one of the effects happens, or the other effect happens, or some third thing happens which is neither the first effect nor the second, but in no case should a logically contradictory state result.
In your explanation both of the example effects have the two damage sources occurring simultaneously and the effects are not so much contradictory as conflicting, so we will from here on deal exclusively with this case.
The exact nature of attack resolution has been a subject of some debate for some time (see this pathfinder question which links to this 3.5 question), and this question to some extent hinges on how atomic the attack action is. While some groups declare that the source of a simultaneous group of effects chooses in what order the effects are resolved, most groups, my own included, declare that all effects in a simultaneous group should be resolved simultaneously. It should be noted that this is a different question, as many groups that play with non-atomic attack actions hold that the dealing of damage is a single atomic step in an attack.
The minority opinion is, in this case, much simpler to adjudicate: The warrior swinging the sword or the mage casting the spell is asked whether he wants the fire or electricity damage to be dealt first (or the Gm just has him declare what the damage dice are from as he rolls if the GM wants to be sneaky, using the order of declaration as the order of resolution). If the electricity damage happens first, the golem is healed and slowed. If the fire damage happens first, the golem is just healed.
The majority opinion is somewhat more complicated since the electricity and fire damage happen at the same time. I personally use the Waving Hands resolution system for simultaneous effects, because I think it makes the most sense and does a good job of being internally consistent. In this case, the electricity effect triggers at the same time as the fire damage is causing the creature to have just begun being slow, so the electricity effect counters the fire one and the creature is just healed by the electricity effect.
Note: This answer is primarily from a 3.5e perspective. Changing this to a 3e perspective doesn't change much so far as I'm aware.
It's not as good a deal as it seems
Bobbi the Builder, a level 13 wizard who possesses the feats Craft Wondrous Item (Player's Handbook 92–3) and Craft Magic Arms and Armor (PH 92), labors for a month in a 500-gp laboratory. During that time, she also succeeds on her Craft (armorsmithing) skill check (DC 20) to sculpt from 500 lbs. of pure iron a magical iron golem limb (Monster Manual II 209). To finish the rituals needed for the limb's creation, she spends 20,000 gp on raw materials and 400 XP, and casts the 5th-level Sor/Wiz spells cloudkill [conj] (PH 210) and 6th-level Sor/Wiz spell geas/quest [ench] (PH 234–5). She puts the limb in her Heward's handy haversack (Dungeon Master's Guide 259) (2,500 gp; 5 lbs.) in case of an emergency.
Attaching the limb
While on an adventure, there's an emergency: Abe the Unlucky reaches into a sphere of annihilation (DMG 279) (minor artifact; 0 lbs.) and, rather than "suck[ing Abe] into the void, [leaving him] gone, and utterly destroyed," the DM rules that only Abe's arm is utterly destroyed. Traumatized, Abe looks at his stump, whimpers, and collapses. (Limb loss in 3.5e is typically at the DM's discretion.)
"I've got this," says Bobbi, and she whips out the iron golem arm. (That she has an arm is enough; handedness, fortunately, isn't really an issue in 3.5e.) She takes a standard action to attach the golem limb where Abe's original limb once was. That's because
Once created, the limbs are treated as spell completion items. Any character capable of casting the appropriate level of spell (see specific descriptions) can attach a limb. [An iron golem limb requires the creature doing to the attaching to be able to cast 6th-level arcane spells, which Bobbi can.] All that’s left to do is perform the final gestures and speak the words needed to imbue the limb with magic. (MM2 209)
This is Abe's first iron golem limb and, after it's attached, his body'll never be able to accept a different kind of golem limb. When the golem limb's attached, Abe makes a Will saving throw (DC 15).
Making the saving throw
Contrary to popular belief, a creature cannot opt to fail saving throws against anything. Creatures can voluntarily fail saving throws against spells (and, by extension, spell-like and supernatural abilities, which are like spells but with exceptions) and a few other similar cases (like psionics) and edge cases (like drugs), but the Will saving throw mandated by the golem limb is neither spell, similar case, nor edge case, so Abe can't opt to fail. (A creature can have effects applied to it to reduce its chance of success on a saving throw, but a 20 will still succeed… except in 3e, where a natural 20 on a saving throw isn't an automatic success nor a natural 1 an automatic failure.)
If Abe succeeds on the Will saving throw, he's gained a pretty significant amount of power asymmetrically: he gains the half-golem template except he retains his own type and gains a +4 bonus to his Constitution score. (And, as the template includes a golem's magic immunity, he'll likely need this power unless the DM allows Abe to lower an immunity voluntarily, an act legal according to one parenthetical example abesent from the SRD on Voluntarily Giving Up a Saving Throw (PH 79) yet nowhere else in entire D&D 3.5e corpus.)
If Abe fails the Will saving throw, the text makes it sound like Abe should become an NPC. That's because failing the saving throw means Abe
becomes a half-golem of neutral evil alignment. The character then has no Constitution score and the character’s type changes to construct, granting him or her construct traits. A neutral evil half-golem retains the memories and knowledge of its former life, but its personality becomes murderous and cruel. It demonstrates [a] hatred of flesh creatures… and it seeks methods appropriate to its class to slaughter as many flesh creatures as possible. (MM2 209–10)
Nonetheless, a DM could still let a player continue playing such a character in an atypical adventuring party. However, the Monster Manual II on Effective Character Level, in part, says
Some of the creatures in this book are capable of having levels in a class, and when they do, they are significantly more powerful than the races described in the Player’s Handbook. This difference in power is expressed as the creature’s level adjustment (a positive number). This number and the creature’s Hit Dice are added to the creature’s class level to determine its effective character level, or ECL. (21)
And, while some creatures in the Monster Manual II do have an ECL, the template half-golem has no general ECL entry, and the sample half-golems—all of them constructs that, apparently, failed the golem limb attachment Will saving throw—from the Monster Manual II (209–12) and its Web enhancement "More Half-golems!" (1–5) don't have an ECL either. They just don't appear to be appropriate for PCs, despite their Advancement entries. (The D&D v.3.5 Accessory Update lists the LA of the Monster Manual II's printed half-golem templates as LA —, and "creatures suitable for use as player characters or as cohort" have an entry that's not LA — according to the Monster Manual (7) for 3.5e. So in 3.5e such creatures really aren't appropriate for PCs.)
Incarnating afterward
Thus this DM assumes that slapping a golem limb on a dude is legit, and the creature, if a PC, remains playable. The downside of magic immunity (if it can't be lowered and, perhaps, even if it can) is such a significant impairment that getting a metric crapton of bonuses is necessary for the PC to continue adventuring at all, and, even then, probably not for long once the novelty wears off. Note that this PC is an ineligible target for the 9th-level Sor/Wiz spell incarnate construct [trans] (Savage Species 67–8).
However, this DM can't imagine a creature that wants "to slaughter as many flesh creatures as possible" agreeing to fail the saving throw against the spell incarnate construct. Okay, this DM can contrive a scenario where that might be a thing ("By taking on a fleshy form, I shall be able to fit in better among the fleshy and slay more of the fleshy! Muhaha!"), but the half-golem's magic immunity (if it can't be lowered) will negate the spell incarnate construct anyway. Yet the existence of the incarnate stone golem (SS 120–1) means someone found a way to cast the spell on that stone golem, so maybe this half-golem could find that caster…
But that's really complicated. In short, a PC could get a golem limb, fail the Will saving throw, and become an NPC. Then that NPC could find a creature who can cast the spell incarnate construct in such a way as to bypass the NPC's magic immunity and become eligible for PC status once more, but the DM must determine that PC's ECL. This DM suggests that this ECL be the PC's Hit Dice −2. The template incarnate construct removes most of the PC's special abilities, including all special attacks and special qualities, which is pretty much everything previously gained from class levels.
Best Answer
Your Interpretation is Correct
In D&D 5e specific beats general. That is:
In this case Damage Immunity: Fire is a general rule that applies to many creatures, allowing them to take no damage from fire. Fire Absorption is given in the monster's statblock and specifically describes how Iron Golems interact with fire damage. This is the more specific rule and the one your should follow:
What about the saving throw?
Immune creatures still make savings throws so the Iron Golem would make a saving throw as normal. It would benefit the Iron Golem to fail this save, however technically they aren't allowed to do this, but many tables allow it (see: Can you choose to fail a saving throw?). Additionally the saving throw will actually be made at Advantage due to the Iron Golem's Magic Resistance feature:
The wording of the fireball spell gives us the order of events:
So first the Golem makes a dexterity saving throw (at advantage), then the damage (8d6) is calculated. On a failed save the target is 'subjected to' the full amount. On a successful save they are subjected to' half as much damage. In either case the Golem takes no damage and instead regains HP equal to the damage they would have taken due to Fire Absorption.
Also note that the order of the saving throw and damage calculation do not actually matter. The damage applied/taken/subjected upon the target occurs after the saving throw is made and Fire Absorption kicks in upon the applied damage.
Regarding the dual listing, my assumption is that the immunity is listed in addition to Fire Absorption to make it less likely DMs will forget about it. I would often only check the Damage Immunities/Resistances section of a stat block, having it there is useful although slightly redundant.