I think the sentences:
are all sentences that ask "How old are you?"
How are they different?
What form is most polite?
いくつ is not normally spelled with kanji. Also, this is actually asking, ‘How many?’ To ask for someone's age, you should use the honorific form: おいくつですか。 This is the standard way.
This is direct, but still in the polite form.
This is direct and also informal. I can't imagine many situations where this would be appropriate. Maybe if you're asking a child what their age is.
I can't say I've ever seen this before...
I will probably go with "お歳をお聞きしてもいいですか？" if I had to ask the age of some stranger or customer etc. If the other person is a female, I might throw in "大変申し訳ありませんが" before the question. It is considered impolite to ask a female her age in Japan. If you have to ask for whatever reason, you are expected to take extra care.
If you are asking an acquaintance, "何歳ですか？" or "何歳でいらっしゃるんですか？" will be more natural. You should use 何歳でいらっしゃるんですか if you think the other person is senior than you. おいくつですか？ おいくつでいらっしゃるんですか is also appropriate in this situation.
年齢 is not commonly used to form questions, perhaps because it is a bit formal or even a bit bureaucratic. A police officer or something could ask 年齢を教えてもらえますか？ ご年齢は？ and so on. It is frequently used in formal situations, as in 国民の平均年齢は or 年齢が高い層の平均貯蓄額はX万円だった etc.
Finally, when you address an young child, you can say なんさい？ or いくつ？.