Forge improvements
There's an awful lot of acrimony regarding the Forge, but I honestly think it is not bad in concept. Execution is another matter though- not only is the UI hard to decipher, but the amount of crystals required to forge jump drastically at level 4 and level 7 to more than double the previous amount, which is extremely discouraging. The changes I propose are:
Make the % success based on amount of HEAT the item has acquired. For heat bar <50%, the success rate should range from 0-40%, the same as the 'Low' crystal value. The 'Mid' crystal value would then correspond to heat bar 50-75%, for success rate between 40-80%; and lastly, the 'High' crystal value would be 75-100% heat bar, with a 80-100% chance. The number of crystals now required at each heat level is constant, eliminating the UI confusion. All it will display is "Number of crystals required" and "Chance of success", with a note saying "To increase the odds of a successful forging, apply more heat your weapon!". The different prizes would also be available to these three levels of heat.
I would also eliminate the level jump disparity- increase the cost of forging each level more gradually. It is all right to require the same number of crystals in the end, but distribute it more continuously so people aren't blindsided by it.

Kat spawn
We all agree that pure random chance can give you lottery stories like 5 kats in a row, but let's face it, a 20+ run drought is extremely frustrating. I propose an entropy-based pseudorandom method to prevent this sort of thing from happening. Here is how it works with spawning Black Kats:
Assume every time a kat spawns, there is a 1% chance it will be replaced by a Black Kat. Under the current system, the game rolls a 100sided die (number between 0 to 1) and if the number is >0.99, the kat becomes a BK. While this means you could theoretically spawn BKs every single run during the event, you can also go the entire event without ever spawning one.
A fairer system is to assign each knight an 'entropy' value between 0 and 1, and each time a kat spawns, generate a random number between 0.005 to 0.015. This number is added to the knight's entropy value, and if that value exceeds 0.99, a BK spawns and the value is reset to zero. This guarantees that if you spawn enough kats, you WILL see BKs. On average, with about 20 kats spawning in a kat heavy level, that means you will nearly always see a BK within 6 levels or so, and at the extreme worst case, every 10 levels.
In a party, one can either use the party leader's entropy value, or be extra generous and allow a kat to spawn any time any party member's entropy value exceeded 0.99. Of course, after a spawn, all party members' values would be reset to zero. In the former case, if a party member's (but not leader) entropy value reached 0.99, it would not reset to zero, as a BK was not spawned. This system actually encourages party play, as overall your odds are now better than 1% (much better, in the latter scenario where anyone's entropy exceeding 0.99 spawns a kat). This system basically enforces the statistical average, preventing normal distribution tail cases. The two main side effects would be the elimination of streaks, both good and bad. I would argue that the guarantee of getting a certain number of tokens for your time is infinitely more appealing than the miniscule chance of getting lots of tokens in a short time.