>>642
1. 初めまして, although 始めまして is not incorrect
per se.
2. Try 'douzo yoroshiku'
As for the kanji, this expression uses only one: どうぞ宜しく
3. お and ご are the same thing. However, their usage differs in that
- お is usually affixed to native Japanese words: お世話, お母さん
- ご is usually affixed to Chinese loanwords: ご意見, ご両親
(As the 'usually' might tell you, there is an exception or two.)
As for your second "kanji+kana versus kana+kanji" question, it's fundamentally flawed. There is no rule on how to use kanji. You can use kanji and kana in any way you want; you can leave the kanji out and write the word in hiragana only, or maybe in katakana if you'd like, or use as many kanji as you can.
And actually, the polite affix お/ご does have a kanji: 御.
Protip: If you've just started learning Japanese, it might not be such a good idea to delve that deeply into kanji yet. Learn your pronunciation, hiragana, katakana, basic grammar and vocabulary first, and then start working on kanji.
It might also help to know that Japanese people themselves usually write all the aforementioned greetings in hiragana.
So, don't fret. Japanese is a difficult language. Don't try to be in all places at once, so to speak, especially during such an early stage in your Japanese studies. がんばって！