Options one, two and four are the same, in terms of English language and usage. They display different typographical conventions. One should be consistent in typographical usage. Stylistically the use of superscript ordinals has decreased as more documents have been created by their authors using typewriters or word-processors. They therefore have a more old-fashioned feel, even when professionally typeset. The italic version is rare in my experience, though it may be part of the style-guidelines of some publications.
In continuous prose it is customary to spell out all numbers up to some maximum. The maximum varies depending on the source of advice, but would include two in all cases. I tend to follow the same practice for ordinal numbers, so would use 'second-order', but '22nd-order'.
None of your options would be unclear, or indicate non-first-language use by themselves. So I'd suggest you chose one you are comfortable with and use it consistently.