You can write both "han klipper snöret" and "han klipper i snöret". For me as a native Swede the first example would mean that the guy is cutting "snöret"(the rope) once, while the second example would mean he is cutting it more than once. The second one also sounds more formal, as you're describing it from an objective point of view. I guess most of it is situational.
Maybe someone more situation into linguistics can explain it better to you if needed.
The more I learn the more explicit the swedish language seems to me. In this example I picture the girl is in the middle of the process of cutting the paper, so you say "klipper i papperet". If you say "klipper papperet" I imagine the girl has just cut the paper but hasn't finished the action of "cutting the paper"(if that makes sense). Is this an accurate understanding of how this might work?
Google search turned up 17 hits for "klipper i håret" but 40,800 hits for plain "klipper håret".
Also got 1 hit each for "klipper i gräset" and "klipper i gräsmattan". There were 3 hits on "klipper i gräs". The same phrases without the 'i' yielded 45,200 , 13,300 and 28,300 hits respectively, including multiple hits that had something to with klipper gräset i bikini and at least one about klipper gräset naken. It was more than likely a dude, I figure.
To cut with a knife is skär. In another thread, someone (I don't remember who, sorry) said that lawns and hair are also klipper. I took it to mean (and I could definitely be wrong) that skär is to cut with a single edge, and klipper is to cut between a pair of edges, although I suppose a lawnmower doesn't quite work that way. We do sometimes say in English that we give the lawn a haircut, though.
Sounds accurate. Swedish Wiktionary defines klippa: "dela eller kapa genom att använda en sax" = (Google Translate) "to split or cut with scissors", which is exactly what "clip" means in English. We cut the grass, but we trim the hedge with a hedge clipper. Wiktionary defines skära as "dela något ting med vasst blad (kniv)" = "to split something with a sharp blade (knife)." When in doubt, I go to a monolingual dictionary.