I never saw even one episode of TVD but shipped Tyler/Caroline and Damon/Elena. And I only watched season 1 of Lost but shipped Sawyer/Kate. I think Tumblr makes shipping couples on shows you don't watch easier since a gifset will appear on your dashboard and you'll be like 'aww, they're cute together!' despite never having seen the show lol.

Oliver/Connor from HTGAWM
Dany/Khal Drogo from GoT
Oliver and Felicity, and Roy and Thea from Arrow
Bellamy/Clarke from The 100 (before I watched it)
Magnus/Alec from Shadowhunters (before I watched it)
sometimes I watch the show just cuz of the couple

Malia+Stiles
Oliver+Laurel
Barry+Iris
Alison+Emily
Jessica+Trish

Very rarely.
Carol and Daryl from The Walking Dead (can't think of any others right now)
I did start watching TVD because I shipped Klaus/Caroline...now I'm regretting ever watching TVD or TO. Both so disappointing, especially as far as ships/great characters go :(

It has happened to me.

Not really. Sometimes I''ll see posts on Tumblr about a certain ship (of a show I haven't started/watched yet) and it'll capture my interest, I'll think they're cute.. but I wouldn't go so far as to say I ship them. I personally don't understand how you can ship characters if you know little to nothing about them or their "relationship", but that's just me.

Rarely, but it happens.

Bamon The vampire diaries :)
Finn and Leah Glee
Matt and Kelly Beverly hills
Liam and Naomi Beverly hills new one