There's no specific confirmation in any of the TV shows, films or official novelisations whether (or not) Romulans have a Katra. Their general lack of telepathic abilities would suggest that they don't have the ability to use them to transfer knowledge, but then again, who knows what goes on at the highest echelons of Romulan society or behind ...
All living beings have a Katra.
But @Valorum's answer is correct - there's no evidence that Romulans are aware of it, or can "use" it.
In Star Trek: Discovery S01E06, Lieutenant Stamets explains his understanding of the Katra concept:
STAMETS: I got to say, this Katra stuff is way cool. An uncharted superhighway connecting all of conciousness and life?
The canon answer (courtesy of The Star Trek Book factbook) is that the Vulcan's interest in Earth was in fostering a relationship with Earth that was to Vulcan's advantage. This included encouraging Earth to progress out into the galaxy at a much slower rate than if they'd supported Earth's technological progress or if they'd gotten out of the way, forcing ...
The goal of First Contact is not necessarily to gain anything, but to avoid the chaos that an accidental encounter could lead to.
In the "modern" Star Trek times, the Federation initiates First Contact with any civilization that is about to develop Warp Drive. There doesn't seem to be any question of "benefit" to the Federation.
There are no concrete plans for book VI. It took the writer some considerable effort to get a contract for book V, and to actually write it once contracted. From his blog
This was kind of a rough one, since there were delays getting the contract and approvals through, so I was late getting started and I had only three months to write the book.
By his own ...
As pointed out by @Joe L, there is no 'canon' on this. I assumed they did this in order to make humanity's interstellar expansion progress more smoothly and prudently than it might have with no guidance, and they did this out of enlightened self-interest - essentially, so that, for example, humans wouldn't give them grief the way Andorians had.
Because they traded technology
From the Memory-Alpha write-up for TOS: The Enterprise Incident
Kirk then inexplicably orders helmsman Sulu to take the Enterprise
across the Romulan Neutral Zone and into Romulan space, where it is
quickly surrounded by the Romulans – who are now using Klingon-style
An effective cloaking device, meanwhile, ...