What if Janeway's dog Mollie had ended up in the Delta Quadrant? I bet Janeway would have been unbearable about it. She would probably insist that everyone salute her, and keep making jokes about how Mollie outranked Harry Kim. Janeway would have all the doors modified to open for her, and she would get Mollie her own com badge and would always be calling her and doing a lot of baby talk and trying to get her to bark. There could have also been a story about how Janeway takes Mollie down to the Airponics bay to crap, and instead of picking it up she just blasts it with her phaser, and it makes the whole place stink. Then Kess would get really upset, and probably everyone would get sick from eating the vegetables. -or maybe The ship starts having problems and they discover that Janeway has been letting Mollie crap in Fair Haven for a long time, and then the Doctor starts to stink because the ships holographic matrix is corrupted, or something. There's also a lot of room for jokes about how Neelix wants to cook her. She could also sense things and get knocked up by aliens like Counselor Troi. I think it would be great.
^Yes. That's how bad the Kazon are Oh, and Molly would have seen Neelix as a prey and B'Elanna as competition.
Molly looked like a big mush, not sure she would have treated Neelix like a rat to be hunted.. oh wait, yeah she would have. Janeway: It's not crunch time yet Molly, put him down!
You forgot to mention how the EMH would be running 24/7 working on life-extension technologies, and that, given a choice between finding the female Caretaker and visiting a race that could give them the biotechnology to extend Mollie's life indefinitely (and only Mollie), Janeway would have agonized over and talked with Da Vinci about it.
But if Mollie had been aboard, we would have had an episode where she had some deadly illness, was put into a 5 gallon pickle jar, and Janeway spent the night in sickbay to keep her company while getting annoyed with the EMH's grooming habits.
I'm not really fond of irish setters, I wouldn't want one galumphing around the corridors. Porthos was excellent.
The Kazon wouldn't have taken over the ship in "Basics" they would of been afraid of Molly and her puppies. The Voth would of commented that the dogs smelled better than the humans in "Distant Origin". Those aliens in "Scientific Method" would have been long detected before the Doc adjusted Seven's ocular implant.
Maybe Mollie would have gotten mixed up with Tuvok in the transporter, and then Janeway would have to make a hard decision about getting Molvok fixed because it's Pon Farr is accelerated by being in dog years.