The question involves trigonometric identities and their implications on triangle geometry. We're given:

1. $A, B, C$ are non-zero acute angles.
2. The equation $\sum (\tan A - \cot A) = \sum \tan A - 2 \sum \cot A$.
3. If the minimum value of $\sum \left(\frac{{\tan^2 A + \cot A}}{2}\right)$ exists, $A, B, C$ will form a specific type of triangle (scalene, right, equilateral, or isosceles).

Step-by-step Solution:

Analyzing the First Condition

Given the identity: $\sum (\tan A - \cot A) = \sum \tan A - 2 \sum \cot A$ This holds true due to the distributive property of summation over subtraction.

Understanding the Second Condition

We analyze: $\sum \left(\frac{{\tan^2 A + \cot A}}{2}\right)$ We need to find the minimum value of this expression to determine the type of triangle. Using the trigonometric identity for $\cot A$, we can rewrite it in terms of $\tan A$: $\cot A = \frac{1}{\tan A}$ Thus: [ \tan^2 A + \