If you cut off your nose to spite your face, you do something rash or silly that ends up making things worse for you, often because you are angry or upset.
This expression 'eye-wash' is generally used to cover up the anxiety of a person who is seeking a concrete reply or justification for an act or an event that had affected his personal image or caused him a loss. The affected person usually represents his case to the higher-ups and puts forth his demands for redressal. But the authority, in order to avoid embarassment to his organisation or to himself, is not in a position to expose the entire material or evidence which in turn tell upon the credibility of the organisation. In such circumstances, he will usually call for an investigation to satisfy the complainant, but will not be keen in disposing the case. The authority will drag on the issue, (at the same time pretending to be serious) until the seriousness of the issue dies down and no finality is reached. So, ' The investigation on the issue by the authority is an eye-wash'.
If you have or get your foot in the door, you start working in a company or organisation at a low level, hoping that you will be able to progress from there.