Wesley creates a hand held tractor beam that can be used to juggle chairs in "The Naked Now," which - when hooked up to the ship's power - can create a large repulsor force field that can stop a human. In "The Battle," we learn that the common cold has been conquered, along with headaches; Ferengi transporters are a few seconds slow. "Only a thousand kilometers" is quite close enough to be put on main screen. "Eye of the Beholder" - if a ship is detected at sub-warp speeds, it must be hours away.
Data is a very curious individual. With the memories of 411 colonists, the equivalent of a supercomputer on tap for linear computations.
They have ten year olds learning calculus in the starship classrooms. Long range transports are different. The captain does orbital physics for fun. People may be placed in stasis.
In "Booby Trap," Data remarks on the debris that once was the planet Orelius 9. He finds the destruction "remarkable considering the primitive weapons of the period."
In "The Enemy," we learn that the cell structure of Romulans, Vulcans, Humans, Betazoids, and Klingons is very similar. A Klingon may donate ribosomes to a Romulan when a number of Vulcans turn out incompatible. "The Chase" further clarifies this relationship; all were seeded by the same ancient race of humanoids.
"Family" - someone wants to build a new subcontinent on Earth. They've been at it for a while..
Tarchannen - home to a very curious species of alien that reproduces by transforming people into it. Chameleons, extremely difficult to detect on all sensor bands.
Picard is an amateur archeologist... who is, in spite of his career, considered a leading expert in some particular topics, giving talks at major archaeological get-togethers. The common archaelogical record of Tagus III consists of expeditions conducted 400-22,000 years ago.
Also in "Disaster:" The antimatter containment field is constructed with a safety factor of 6.67 - there is no danger of containment breach unless containment fields drop below 15%.
"Realm of Fear" - individuals remain conscious during transport - including between sites.