Indeed, seaQuest DSV has often been called "Star Trek, underwater".
Consider the following remark in an IGN retrospective on seaQuest DSV :
[seaQuest DSV] sailed through a three season run and after its cancellation, it didn't have the legs in reruns as did Star Trek: The Next Generation and DS9, the very programs that inspired it.
However, this claim is not substantiated in any way by the article's authors. The question is: was DSV really inspired by the various Star Trek series, or are any similarities unintentional?
In addressing the question, I'll go from the weakest points to the strongest points.
"DSV" from "DS9"?
There is no publicly-revealed information that suggests that the acronym "DSV" was influenced in any way by "DS9". The acronym "DSV" for deep-submergence vehicle has a long history, having been used in naval circles since at least 1960. Its selling point as a title is that the acronym rolls off the tongue easily and sounds "cool" — and that's probably why Rockne S. O'Bannon (the creator of seaQuest DSV) chose it.
Since seaQuest DSV started airing only nine months after Deep Space Nine, it is hard to assess which name was conceived of first. But all in all, the fact that "DSV" was already a well-established real-life acronym casts doubt on a link between "DSV" and "DS9".
"seaQuest" from "Star Trek"?
This has never been confirmed or denied, but it does seem reasonable. (I'm not saying that this is a weak point in and of itself — only that there is no confirmed statement to this effect. I agree that there is little debate.)
Of course, DSV was squarely in the sci-fi genre, and as such it had sci-fi adventures. As for whether the particular adventures on DSV are Star Trek-esque, a case can be made for this. This has to do with the direction the show began to take after its first season, and also with the audience it was trying to court.
In Season 1, seaQuest DSV focused on, well, the sea — as it should. Come Season 2, a large focus was placed on aliens hiding on Earth. There was also an attempt to lure Star Trek and Star Wars fans over to DSV by having Mark Hamill and William Shatner as guest stars.
For example, Mark Hamill playing a man who is actually an alien hiding amongst humans, who is then pursued by a member of his own kind who wants to take him back to his home planet to face sentencing and execution for tenuous crimes, is reminiscent of several actual Star Trek plots.
What the show can be accused of with definite certainly is having an identity crisis. It was never sure of what it wanted to be, and this only worsened as it went into its second and third seasons.
Ensemble cast on a ship with a young, nerdy, annoying know-it-all
I've combined three of your points into one. As others have pointed out in comments below your question, the ensemble idea is not exclusive to Star Trek, although The Original Series may have perfected the concept. The idea of a crew on a ship, be it the Enterprise or the Galactica, is probably so ingrained in public consciousness that it would have taken the creators some effort not to set up the show in that way.
One might say that Lukas, the whiz kid on DSV, was a nod to Wesley Crusher, but then again, television has always had such "boy geniuses", from Doogie Howser M.D. to the kid on My Secret Identity. (The trope also appears in the La Femme Nikita ensemble-cast television series with its character Birkhof, around the same time as DSV.)
As for the inspirations behind other characters:
Roy Scheider's character was based on John C. Lilly. Lilly was a pioneer researcher into the nature of consciousness using as his principal tools the isolation tank, dolphin communication and psychedelic drugs, sometimes in combination. He was a prominent member of the Californian counterculture of scientists, mystics and thinkers that arose in the late 1960s and early 1970s. Albert Hofmann, Gregory Bateson, Ram Dass, Timothy Leary, Werner Erhard, and Richard Feynman were all frequent visitors to his home. The character's name, Nathan Hale Bridger, was in homage to Nathan Hale.
When producers began developing new characters for the second season, they named Lieutenant Brody after Police Chief Martin Brody, Roy Scheider's character in the first two Jaws films.
References to Star Trek (and vice-versa)
Whatever the case is regarding the inspiration behind seaQuest DSV, the show did intentionally reference Star Trek at some points. Consider the text in this screenshot, for instance:
Also this, from the episode "Dream Weaver":
During DSV's second season, TNG returned the favour. In the episode "Eye of the Beholder", the following character's service history includes time on a ship called the "Seaquest":
seaQuest DSV's attempt to carve out its own identity in spite of its similarities with Star Trek is discussed in the following article: