There was the Pon Farr episode on the original ST series where his promised Vulcan wife didn’t want Spock in part (IIRC) because he had become something of celebrity. Was this notion of Spock as being a big deal on Vulcan ever developed further in any of the Trek Series.
In TOS, only implicitly. Spock’s father, Sarek, was stated to be a Big Deal ambassador. And Spock was stated to be the first Vulcan in Starfleet. Those two things in combination would, it stands to reason, confer extraordinary recognition on Spock.
In TNG, the two part episode “Unification” implied that Spock was a household name in the Federation (or at least in Starfleet), and also that he had become a well-known ambassador in his own right.
Not to mention the fact that Spock is one of the first, if not the very first, Vulcan Human hybrids. That all by itself would make him something of a celebrity. Remember Louise Brown?
The often overlooked connection between Spock and JFK.
There were rumors to the effect that Sarek had certain “business interests” in Romulan Ale, as well as an inexplicably large fleet of cargo vessels.
This is something that struck me as odd. In First Contact we find that the Vulcans contacted us, having well developed warp travel. They must have had star travel for a long time, since a branch went to Romulus. So, why was it a big deal that Spock went to StarFleet? It’s like an American from a famous family going to the military academy of a third world country - interesting, but not worthy of celebrity status.
I couldn’t stand watching Enterprise after the first few shows - did it explain how
Earth got to be such hot shots in a short period of time?
Yes, but only in the final three seasons, which never aired in the present version of your timeline. Damn Temporal Cold War!
There was also the geneological connection to one Richard Grayson of Gotham City, through Spock’s mother. 
This is widely bandied about, but clearly not true. Was it actually stated in the show or did this come from one of the novels?
I believe the current fan accepted explanation (or at least most widely) is that he was the first Vulcan to attend Starfleet Academy.
Vulcan effectively became part of the Federation; they didn’t really colonise anything much (besides Romulus and a few other planets), while the Federation grew in leaps and bounds. After a couple of hundred years, the patron/protegee relationship between the two civilizations was effectively reversed.
Think of it as a Briton immigrating to the US in today’s world.
Spock was chosen to announce the Romulan annexation of Vulcan in Unification, a two-part TNG episode, so presumably he was considered an instantly recognisable figure to the average Vulcan.
Look, the Wayne interests have been trying to hush up that sordid story for years! 
This isn’t canon, but in Diane Duane’s novel Spock’s World, T’Pring is shown to have been an extremely successful and moderately power-hungry politician. Had she been Spock’s Wife rather than T’Pring, living in his shadow would probably have held her back.
I believe this novel also detailed the history of the Vulcan/Romulan split and the Romulan Exodus, which was accomplished using incredible psychic powers which the Romulans were okay with, but the Vulcans weren’t, which is why the Romulans left. Again, not canon.
Ironically, the psionic-space travel involved a psychic jump to light speed which tended to kill the psychic, so by the time the Romulans got to Romulus (or whatever it was called…), they were fresh out of psychics, which is why Romulans have little to no psychic powers–they bred it out of their bloodlines by killing them all off, whereas the Vulcans still have it, but only in a very limited way. Again, not canon MK III.
That explains why the Romulans didn’t have warp capability even at the time of Balance of Terror, even though they would up a long way from Vulcan. Oh what a tangled web we weave …
Ethilrist, the Romulans’ name for their planets are ch’Rihan (Romulus) and ch’Havran (Remus). Their name for themselves is Rihannsu.
Spock’s World was one of my favorite books, and the Romulans easily my most favorite species.