1) We see Klingon tractors in e.g. "Way of the Warrior", and they look just like the Fed ones. We hear of Romulan tractors in e.g. "Improbable Cause", although the visual effect is not seen IIRC. We see Cardassian tractors in e.g. "Dax", as the station DS9 is of Cardassian manufacture and supposedly so are its tractor beams; we also hear of Cardassian shipboard tractors in "The Maquis".
We haven't yet specifically heard of any TNG era culture specifically and explicitly lacking tractor beam technology if it possesses things like warp drive, shields and death rays. In the ENT era, lack of tractors is a sign of backwardness for mankind, and several competitors such as Vulcans and Klingons already have those.
2) Shielded targets can be tractored with a powerful enough beam, but shields do make the ship "slippery". The Borg are notorious for catching shielded ships and then proceeding to drain the shields, possibly with the help of a special tractor beam (but regular tractors don't necessarily have a shield-draining quality). Our regular heroes or villains seldom manage to slip from a tractor beam, shields or no shields.
3) We haven't heard of a tractor tearing or shaking a starship in half deliberately, but weak objects may be torn to pieces by a single beam, as in "Tomorrow is Yesterday" where a F-104 fighter was disintegrated by a beam. Also, in "Paradise", a warp tractoring is said to pose the risk of both the involved ships being torn apart.
4) Repelling an entire if unresisting Oberth
class starship was done in "Naked Now"; partially repelling a huge Klingon attack cruiser in combat conditions was done in "Way of the Warrior" (or at least the beam repelled the disruptor fire from that ship - we don't know if by affecting the beam, or by shaking the ship itself to disrupt the aim). Both took place at point-blank ranges. Pushing a major stellar body was attempted in "Paradise Syndrome", but with deflectors rather than tractor beams - yet a planetside device acting like a tractor beam ultimately deflected this major, almost Moon-sized body. Similarly, in "Demons", it was told that United Earth in the 22nd century employed a tractor-like planetside device capable of deflecting comets at interplanetary distances.
5) Starship tractors have only ever been used at close ranges. I don't remember references to specific maximum ranges, but I have the nagging feeling there might have been some. The planetside system from "Demons" did have considerable range, easily reaching from Mars to Earth.
6) Speculation only... Tractor beams might add to the destructive potential of a starship, but apparently only at close ranges. Probably they are more trouble than worth both because other types of weapon will decide the battle before that close range is reached, and because creating a tractor connection between two maneuvering vessels poses some risk to the tractoring vessel as well.