Getting to the crossover thing, it really does depend on how the studios want to handle it.
NCIS & NCIS LA have regular cross-over with a single character (Vance) appearing in both - pretty much has to be the case as Vance is playing the NCIS Director, so would start to look odd if he's always referred to off-screen! But others pretty much haven't, the whole cast of LA in an NCIS two-parter which was the pilot anyway, then Abby in one LA episode and that's been it.
CSI/CSI Miami/CSI New York have had a few cross-overs, even a three-part story where one part took place in each show. But once again, not many.
Even Hercules and Xena had few cross-overs, Ares and Autolycus crossed over, Hercules appeared in a couple of Xena episodes and vice-versa, with actors sometimes portraying other characters (Yes Virginia, There Is A Hercules
But Trek itself has usually kept out of the cross-over game, apart from the noted exception of Pilots. Encounter at Farpoint
, and Caretaker
all featured a character from the prior show, to indicate to the audience that this is "real Trek".
They still fitted in a few such as Kurn, Toral, Thomas Riker, Kang/Kor/Koloth, Barclay and Troi, for example, but I think on the whole the powers simply wanted the different shows to be their own animal. Had they wanted to mention Picard as being more involved in the Dominion War, or had a multi-season spanning arc featuring Thomas Riker taking the Eddington Role in the Maquis, or have DS9 mention that they were following up leads in the badlands that may explain the fate of USS Voyager
, they would have.