Please take a look at the Icicles documentation. Section Icicles - Ido and IswitchB covers this.
It tells you, for example, "You cannot use Icicles and Ido together, however – they use the minibuffer in incompatible ways."
However, you can use Icicles and Icomplete together. Icomplete shows you completion candidates in the minibuffer (and lets you choose them) in a way similar to Ido.
Icicles - Ido and IswitchB also tells you how you can get more Ido-like behavior while using Icicles. There are different parts of Ido behavior that you can obtain, with respect to: incremental completion, non-expansion of minibuffer input, matching, cycling and sorting, current candidate, editing input, display of completions, auto-choice of single candidate, and more.
That page also tells you that to get Ido-like behavior for files (your
C-x C-f example) you just need to customize option icicle-files-ido-like-flag.