DK: I am able to see newly created form in nested form . How to make new form appear in below dropdown?

If I've got you right it will require the AJAX call. How it works - in your custom module PHP code) you have the method that generates the forms list. In the form you have the Javascript / JQuery command loading the mentioned method from PHP.

DK: Where I can write code to do validation can you provide any example?

Every form field in the Developer App has the "Check for" parameter where you may choose the variant for validation, for the date and time fields it looks like on the pic below:

Or you may set the field as the text and write there the Regular expression check like:

DK: How to make view form displayed on profile? Which block I need to add on Profile Info page and how to define which form result to be printed?

The easiest way is to add the second Profile Info block and hide the unnecessary fields with the CSS.