We don't know enough about him yet. He might have other traits not yet shown.
He also had Vulcan pon farr urges like a pure Vulcan and those emotional responses due to he human heritage caused him not to act like a pure Vulcan. That's why he had issues on Vulcan. He also survived a blood draining creature, because his blood isn't completely like a human, but still was affected to a degree, because he is part human.
As you note, the Doctor is a bit of an oddball.
Except that we know for a fact that Spock did, because it came up. And we don't know enough of Jacen to determine that he doesn't.
As for mixed features in fiction...
Wil Ohmsford who was more human than elf, but still was able to use the elfstones due to that bit of elven heritage.
Every changeling in The Dresden Files. Unless they go fully fey, they are a pretty even mix of human and fey with abilities in the middle.
Fleur Delacour from Harry Potter was mixed human and veela and had more abilities than human, but less than a full veela.
As you note, all the half-elves in Middle Earth, as well as the Dunedain and Numenorians who had powers such as mind speech due to their non-human bloodline.
Calis from Midkemia(Raymond Feist is the author) has an elven and a valheru parent and is more than an elf, but less than a full valheru in ability and power.
In David Drake's Lord of the Isles there a few human/other(can't remember the names) individuals with power beyond human, but less than the parent.
Neil Gaiman in Anansi Boys and American Gods. The scions of god and man are more than human, but less than full god with their abilities.
L.E. Modesitt in the Order War series. Those with magical powers have mixed blood from their ancestors, but not as strong because it's not pure.
Blade.
I could find a lot more if I tried.
Edit: All the half-gods from Percy and the Olympians. Hagrid. Luthien(half elf/half-maia). Aquaman who could breath in air and in water, unlike both parents, and was stronger than the other Atlanteans. Hellboy. Sisko who was half Prophet. More and just keep popping to mind.