Standard (pdf)LaTeX based on pdftex is 8-bit orientated.
If you use an 8-bit encoding like ansinew, ä is encoded as 11100100. This means it is 8 bits long and so for pdftex it's a single input "entity". In this case \^ä can work (if you use inputenc, ä is a command and what happens depends on the font encoding which sets the actual definition of ä).
If you use utf8, ä is normally encoded as 1100001110100100. So it is at least 2x8 bits long and pdftex sees two input "entities". As inputenc makes the first bit active and so ä is actually a rather complicated command, a combination like \^ä will normally break.
The engines XeTeX and LuaTeX are unicode orientated.
For them a UTF-8 encoded U+00E4 ä is a single entity. With these engines there is no longer a difference between "normal" ASCII and "special" non-ASCII chars: ä will be handled like an a and \^ä will work fine. If the ä is encoded as U+0061 U+0308 then there are also for these engines two input entities. A command like \^ä will normally not break in such a case but the output could be different from \^{ä}. Again what happens depends on font and on the actual definition of \^ (normally set by xunicode with these engines).
xunicode(maybe viafontspec), there's nothing special in the characters the engines see. And of course, they won't be printed in most cases, as they correspond to inexistent font positions. – egreg Jul 18 '12 at 09:24