sware - "to answer"
sweger - "mother in law"
sweor - "father in law"
There are many. Often the native word coexists with the borrowed Norse cognate, as in yard / garden or shirt / skirt.
give is, I believe, subject to some debate. Without Norse influence, it would be pronounced yiv. People argue over whether it should be thought of as a borrowing from Norse or as a reversion of the pronunciation of the English word in a Norse-heavy environment. (etymonline has the second of those theories; wiktionary has the first.)
No, Old Norse is Old Norse.
"Germanic" (i guess you mean Proto-Germanic) and Old Norse are both indo-european languages.
Nordic languages (a family of languages) are descended from Proto-Nordic (a language).
Germanic languages (a family of languages) are descended from Proto-Germanic (a language).
Germanic languages are Indo-European languages, which have Proto-Indo-European as a root language.
You are not wrong in saying that Proto-Germanic and Old-Norse are IE languages. But it's also like saying that nickles and coins are currency, it doesn't mean that nickels are not coins.
My point that got lost along the way was about Norse ruling britain for 200 years and their cultural influence on the old english during that time.
They come from Indo-European roots and are older than Old Norse or Old English. Cognates are also found in Slavic and Roman languages:
sware - свара - sermone
sweor/sweger - свекор/свекровь - suocero/suocera