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
<p>みど理 </p>
<p>みど里 </p>
<p>み外里 </p>
<p>み登り </p>
3 Kanji compounds:
三十里
<p>三都里</p>
<p>光巴里</p>
<p>光都里</p>
<p>妙登利</p>
2 Kanji compounds:
三彩
<p>光鳥</p>
<p>実酉</p>
<p>常緑</p>
<p>碧里</p>
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"