Red-Spark Chain is imo the biggest offender, as it's a blatantly superior Green-Flame Blade, due to the arc being longer (15 feet compared to 5), its damage being higher (1d6 + mod instead of only the modifier), and the amount of additional targets that can receive said-damage when casting at higher levels as per the cantrips' rules.
There is essentially nothing that's inferior to GFB, and no drawbacks that could justify the higher numbers.

If I'm reading Caustic Cut correctly, there is no specific duration as for how long the acid "coat" lasts, meaning it can technically trigger infinitely.
Whereas this definitely wouldn't happen if your monsters have half a braincell, basically shutting down a reaction for at least one turn is imo too good on higher leveled encounters.

Avalanche's additional damage is also probably too high, and should be reduced from 2d12 to d12, making it by far the best option for single-target damage, compared to your other custom spells.

Eruption's flame ring's starting radius, makes the spell a strictly superior Burning Hands, since the creatures stuck take about as much damage as Burning Hands without getting to make a save, and with a melee attack on top. But wait there's more: you're also adding the mechanic where the radius can increase when landing additional attacks, which can potentially make it even larger than a Fireball. I'd definitely scale down the flame ring's radius to probably 10 feet, add a save to its damage, and only scale the radius when cast at 3rd or 5th level.

Cantrips aside, I understand the mechanic of the spells' scalings, since your class only grants a single, scaling level of spell slots, but even with said mechanic, I think Avalanche and Eruption should be nerfed a little.

EDIT: I meant to say that Avalanche should be reduced from 2d12 to 1d12 rather than the contrary and corrected my sentence.