Adding on to oals' answer:
to finish; to close; to do something completely; to put away; to put an end to
Common word, Godan verb with u ending, Transitive verb, Usually written using kana alone
Source: edict, searchable on jisho.org
It seems only しまう and 仕舞う are in common use (the others being rather obscure), and even among those two, I've only ever come across the hiragana form.
While not a native speaker, they do all seem to refer to the same word. Any difference in nuance would come from the spelling (kanji is generally perceived to be more professional, etc..)
Also, all of the spellings you listed are listed within the same entry in both the 大辞林 and デジタル大辞泉.
I don't believe しまう is used very often aside from the common ~てしまう construct. There are many other alternative and more commonly used words used depending on the context. (終わる, 終える, 成し遂げる, and so on..)
There is also the expression しまった, which you may see.
Regarding your second question, if you are intent on using the verb しまう, I would stick to しまう as the other spellings are not in common use (as denoted on デジタル大辞泉 with ▽).
Correction from DariusJahandarie:
「しまう」 is used commonly as a regular verb, and even in those cases it's almost never written using kanji (although using kanji is more acceptable here than when it's being used as a 補助動詞).