Possibly the 1980 Dragonstar series by David Bischoff and Thomas Monteleone?
Points that match are that it has a long spaceship habitat serving as a DNA terrarium of reptilian and other types of life. The scene with a T-Rex being created and killing someone occurs in both the first two books of the series. An excerpt from the first book is below along with the Covers of the books if any of those look familiar.
I don't remember the main character liking cats. At least not in the first book of the series.
Day of the Dragonstar
Book One of the Dragonstar Series
by David Bischoff and Thomas F. Monteleone
An artificial Jurassic world spins through space, kilometers long and made to order, build to endure for an eternity. An enticing mystery for the humans of the Heinlein exploratory mission, it holds deadly secrets from the universe's savage past.
INSIDE THE MILES-LONG SPACESHIP IT WAS 160 MILLION YEARS AGO ...
Day after artificial day, outwitting the carnivorous saurians that had devoured their shipmates, the two survivors of the Heinlein expedition to the mysterious object known as Artifact One picked their way through the vast, horizonless jungle that filled the hull of the star-traveling terrerium.
They did what a man and a woman fighting together to survive usually do:
They prayed for rescue.
They searched for a way to escape.
They fell in love
Then, from a rise in the forest, they saw a wall.
And something on the wall saw them.
Atop a crumbling rampart stood a sentry, weapon grasped in its four-fingered hand, partly clothed ... its snout instintively twitching at the scent of the man and woman walking toward its city.