Searching on a name dictionary you'll get a long long list (93) of "midori" as a girl's given name. This excludes "midori" being used as a family name or a place name.
"Midori" is not limited to the kanji for green though. It can be made up of other kanji having 名乗り
(nanori - name reading) of "mi", "do", "ri", "mido", "dori" compounded to form "midori".
And yes you can use 緑
or 翠
as a standalone kanji for the name Midori.
For brevity I will not list all 93:
Kana and Kanji mix:
みど梨
みど理
みど里
み外里
み登り
3 Kanji compounds:
三十里
三都里
光巴里
光都里
妙登利
2 Kanji compounds:
三彩
光鳥
実酉
常緑
碧里
There is also the possibility that a name is spelled purely in Hiragana:
みどり
In this case, writing that person's name using Kanji would be wrong.
For Midorino:
緑野 [みどりの]
(Don't worry about the meaning of "no" it's just the way the name sounds) it can be both a girl's name and a family name.
緑埜 [みどりの]
is a family name instead of a girl's name.
Alternatively it could be Midori + Genitive case marker の.
i.e. みどりの本
to mean "Midori's book"
or 緑の本
to mean "a green (coloured) book"