You can have a look at Ingrid Bauer's paper

Bauer, I., The classification of surfaces in $\mathbb{P}^5$ having few trisecants, Rend. Semin. Mat., Torino 56, No. 1, 1-20 (1998). ZBL0965.14029.

You can have a look at Ingrid Bauer's paper

Bauer, I., The classification of surfaces in $\mathbb{P}^5$ having few trisecants, Rend. Semin. Mat., Torino 56, No. 1, 1-20 (1998). ZBL0965.14029.

It turns out that, if a smooth surface $X \subset \mathbb{P}^5$ satisfies $\mathcal{T}$, then $\deg X \leq 10$. Moreover, a fine classification of these surfaces is provided (they belong to eight families).

As a consequence, if $\deg X \geq 11$ then $\mathcal{T}$ does not hold (and so the answer to your first question is negative).