You won't find Kirk, Spock, Picard or Sisco in SFB. The Star Fleet Universe is a parallel dimension to Star Trek. If any of those Star Trek luminaries found themselves on a SFB's version of a Federation Heavy Cruiser, they'd probably be able to find their way around, no trouble. They'd just be wondering what happened to the Ferrengi, Cardassians and Borg and who the heck the Lyrans, Kzinti, Hydrans and ISC are. They also wouldn't recognize the Klingons. The ships would look the same, but SFB Klingons are still the swarthy, smooth-skulled, sneaky, back-stabbing "Mongol hordes with spaceships and ray guns" (thank you, David Gerrold) you love to hate from Classic Trek. Also their ships don't cloak. The Classic Trek time period corresponds to SFB's "Middle Years", Y155 or so (the 20th century is like Y -50, I think).
Being a wargame, SFB's history is a tad more violent than Trek-proper's. In point of fact, SFB has more violence than you can shake a phaser at. In addition to the expected alliances and tensions, Orion Pirates plunder ships in every corner of known space. Space Dragons pillage planets flying on warp-driven wings while dangerous toys from dead alien civilizations roam the darkness between stars just waiting for something to destroy. Just about every race has has a war, incident or misunderstanding with about every other race. Which, of course, makes for interesting combat situations, and that's what the game is all about.
The core time period for the game is the General War, which engulfed known space for 15 years starting with Y167. Earlier time periods, players sometimes feel they have to "make do" with more limited ships. The General War saw standard starship technology reach a zenith of power and efficiency before even as cutting-edge X-technology appeared on the scene.
As the General War was concluding (with the Federation and its allies victorious) A race of peace-loving people (the InterStellar Concordium or "ISC") decided to save this part of the galaxy from itself. They believed that by the time any civilization went to the stars it had to have mastered the important lessons of peace and non-violence. To them, the races of SFB (even the Federation) were all dangerously psychotic. Having no alternative, the ISC laid plans to build their own fleet of warships to contain the insanity. In Y185, the ISC instituted a "Pacification Program" keep the warring races apart and stop the madness. It even worked. For a while. 15 years of constant warfare had exhausted the economies across Known Space, and the ISC were fresh and had built their fleets for the job. The ISC Pacification Program gave the Star Fleet Universe a badly-needed breather, which turned out to be a lucky break. None of the other races appreciated the ISC's selfless service. Contained by the ISC, the other races had time to rebuild their economies and their war fleets readying to break the Pacification Program.
Then the Andromedans invaded.
The Andromedans were a mysterious race with disorienting technology from another galaxy whose ships had made "Flying Duchman"-like appearances occasionally for several decades. Nobody has ever seen an actual Andromedan and lived to tell of it. The Andromedans appeared in force in about Y192. It seemed like they were everywhere at once. Every race in known space, even the ISC, had to cooperate against the common enemy or all would be conquered. This culminated in Operation Unity, a three-pronged assault that destroyed the Andromedan staging area in the Greater Magellanic Cloud. Some Andromedan ships escaped destruction, but the Andromedan Invasion was over and the era of X-ships had begun.
SFB X-ships very roughly correspond to the ships of the first 6 Star Trek movies (the ADB didn't actually license anything from the movies, so this is an approximation). In SFB, they started to appear in the last years of the General War, starting in Y180. X-ships were a revolutionary "new breed" of starship. They were the first fruits of some of the advanced research done during the General War. They were faster, carried more weapons and were highly sophisticated. Not much has been done with X-ships beyond the original "X1" Module released 8-10 years ago, but this may be changing since a five-year-long running discussion on advanced technology got going on the ADB's discussion boards.
If there's sufficient interest, I'll publish a more-complete history of the Star Fleet Universe (ADB willing, of course).
SFB has no rights to material from the Trek movies, Next Generation or any of the TV shows that compose the "Trek Franchise". At this point, I think it would be awkward to add this material anyway. Over the last 20 years, SFB has grown and changed, evolved into its own thing. Adding Next Gen material to it would dilute that individuality. Since Paramount treats SFB as an ugly stepchild, there's little danger of this happening. That hasn't stopped a lot of individuals from playing around with Franchise ships adapted to SFB. The last time I googled, I think DS9's Defiant has like 5 or 6 different versions out there.
About Play-by-E-mail SFB...
Good SFB opponents can be hard to find. Sometimes a SFB aficionado can't find *anybody* near where he lives. I have seen at least two systems for playing over postal-service-mail, the first dating to the mid-80's. I played a few SFB games this way in the mid-90's. With the development of the Internet, people have begun to play SFB over e-mail. Jim Hart, Steve Rushing and Sandy Hemenway started a SFB tournament on CompuServe and it has mushroomed beyond all reason. Enter one naive Vorlon stage right.
Update: With the development of a subscription service called SFB On-Line by the ADB, interest in play-by-e-mail has waned. SFBOL can run a duel in a few nights whereas the same thing on PBEM can take 6 months. There's still a decent crowd who likes PBEM and I'm one of them.