Whatever you put in the Subject line of an email should contain a brief summary and be relevant to whatever is in the body.
In your examples
Regarding application for the post of product manager
Regarding scanned copy of my ID card
one would expect the Subject lines to be instead
Application for the post of product manager
Scanned copy of my ID card
The Subject line can be made even more specific by adding *your name followed by a '-'(hyphen) and the rest of your planned Subject line
John Doe - Application for the post of product manager
Jane Doe - Scanned copy of my ID card
int his way it is easier for the recipient to search for you without needing to remember you remain address.
Most email programs will prepend RE: meaning in regards to or in response to when replying to an email. The response Subject line would then look like this
RE: Application for the post of product manager
RE: Scanned copy of my ID card
Using regarding in your example is not necessary since the email is obviously regarding whatever is also mentioned in the Subject line. It is usually better to be less wordy as with a headline