Unicode domain labels have an ASCII form for DNS use. RFC 5890 calls the Unicode form a U-label and the encoded form an A-label; the familiar xn-- prefix identifies that ASCII-compatible form.

No. A registry’s IDN table defines the characters it makes available and may have separate tables for languages or scripts. Check the extension’s rules and its registry table before choosing a name.

No. Characters can look alike while representing different domain strings. Registries define any variant-character rules in their IDN tables, so a visually similar name is not a substitute for checking the specific name you intend to use.

Yes. Search accepts valid Unicode domain input and converts it to the DNS-compatible form needed to match an extension. That conversion checks the form of the name; it does not decide the registry’s character policy.