On the other hand, the Tholians had already demonstrated the ability to destroy the Enterprise
from a distance with their death rays. It thus seems rather possible that the Tholians were simply snaring the Enterprise
in a trawl that would allow them to drag her away, back to their base.
In the ENT episode "In a Mirror, Darkly", we see a web that could be used as a weapon: it's deployed much faster and can englobe even a somewhat maneuverable ship, eventually crushing it. The TOS web doesn't seem capable of that under any circumstances, and IMHO shouldn't be considered a weapon.
As for dodging incoming fire by moving up or down, everything in Star Trek of all eras points to this being best done by turning your bow towards the direction you want to go to. Starships don't move sideways or up or down any more than cars or aeroplanes do, except at a crawling pace. And you definitely don't want to crawl in the transverse direction if a torpedo is heading towards you.