WoTC have released an Unearthed Arcana Wizard subclass called Lore Master. One of the features this subclass gives is:

Spell Secrets

At 2nd level, you master the first in a series of arcane secrets uncovered by your extensive studies.

When you cast a spell with a spell slot and the spell deals acid, cold, fire, force, lightning, necrotic, radiant, or thunder damage, you can substitute that damage type with one other type from that list (you can change only one damage type per casting of a spell). You replace one energy type for another by altering the spell’s formula as you cast it.

When you cast a spell with a spell slot and the spell requires a saving throw, you can change the saving throw from one ability score to another of your choice. Once you change a saving throw in this way, you can’t do so again until you finish a short or long rest.

This seems to cover both effects that you wish your Mage type character to have.

So WoTC appear to be looking at the same sort of concept that you want for your homebrew campaign.

Looking at this trait however it is evident that WoTC, at least, consider the changing of damage type to be not that big of a deal to game balance, but the changing of the type of saving throw to have a potentially greater effect (and thus needed to be restricted).

Taking your fireball example, changing the saving throw from DEX to anything else would eliminate the rogue’s evasion class feature.

In general the saving throws have been sculpted to the type of effect the spell is imposing.

Some salient examples are:

• area of effect spells are largely DEX saves (fireball)
• a spell effecting a targets mental faculties are usually WIS or INT saves (Fear or Feeblemind)

Important Caveat

It obviously comes with the caveat that this subclass hasn’t been playtested as extensively as the subclasses published in the books, and as a result is not necessarily balanced within the game.

The Fire Elemental would take more damage.

The water elemental is not immune or resistant to fire damage. This would indicate that the fire elemental's fire form ability would damage the water elemental. The fire form ability can inflict damage in 2 different ways:

1. A creature that touches the elemental or hits it with a melee attack while within 5 feet of it takes 5 (1d10) fire damage.

This would indicate the water elemental would take 1d10 damage if it enters the fire elemental's space, because they would be touching.

2. When the fire elemental enters a hostile creature's space and stops there, the fire form ability has three separate parts:
   • the creature takes 5 (1d10) fire damage.
   • the creature catches fire
   • until someone takes an action to douse the fire, the creature takes 5 (1d10) fire damage at the start of each of its turns.

In #2 the first part is separate from the second two parts. There is no indication that catching fire causes the initial damage, it is presumably high levels of heat. However there is very clear indication that catching fire causes the additional damage on new turns (until someone takes an action to douse the fire...). And because it indicates that it can simply be doused, we can safely presume that it is non-magical fire that will be immediately doused by the water elemental's form.

So the water elemental would take the initial damage, but none thereafter.

However the water susceptibility trait of the fire elemental would also cause the fire elemental to take damage.

Water Susceptibility. For every 5 feet the elemental moves in water, or for every gallon of water splashed on it, it takes 1 cold damage.

In both cases above the two elementals are in the same space and touching, and I would consider that the same as splashing. It would be up to the DM to determine how much damage the fire elemental takes, but it would seem to me that it would be a lot more damage than the fire elemental would cause to the water elemental. A water elemental is listed as Large and I would guess made of at least 100 gallons, that's up to 100 hp damage to the fire elemental. The fire elemental wouldn't get all that damage, maybe 1 - 3 d10 depending on the situation.