I assume gre is so heavily associated with hair that gre seg is fine. Like in english we could say 'he does not floss'. No need to say floss between teeth because where else would you floss? Where as with comb in English we even comb beaches. 'Han likke gre strander' would surely be ridiculous?
It doesn't look like there's any connection.
Groom comes from the Middle English word grom or grome meaning boy or youth. Middle Dutch, Old Icelandic, and Old French all had similar words all generally meaning boy or manservant. The root of all these words is uncertain, but is thought to possibly be the Proto-Germanic word grōmô which is related to the word grōaną meaning to grow. I can't say for sure, but I would have to guess that the jump from boy/manservant to groom came from the boys that were in charge of brushing the horses in the stables.
I couldn't find much on the etymology of å gre but according to the Norwegian Wiktionary page for the verb, an alternate form of it is å greie, which has multiple meanings but to comb one's hair is one. This goes back to the Old Norse word greiða which could mean to comb, and then further back to the Porto-Germanic word garaidijaną which meant to set in order/arrange.
All of this was taken from Wiktionary, if you're interested in looking into it yourself.
Well, yes and no. Google is great with frequently translated language pairs, since there has probably been a lot of feedback. It's usually accurate from Spanish to English and vice versa, for instance. Although you can get some weird translations, in an example that happened to me you get a dip made of hummingbird instead of a hummingbird dipping. (a poem with the line the hummingbird dips) On the other hand - even with single words it can be dicey if it's not a commonly looked up language. I'd never put more than one single simple sentence in without having a bit of an idea what it should be, though.
In NOR this sentence would normally mean that he does not like to comb his hair.
Obviously a smart fellow. http://www.thelifecoach.com/wp-content/uploads/2013/01/AlbertEinstein.jpg