Can we use double quotes for strings inside t()?
A string is a string... PhP allows strings to be defined four ways, but the two that apply here are 'xxx' and "xxx"... both are valid strings that will be passed to the t() function.  The function cannot determine how the string variable was created, it only gets the content between the delimiters.  So you can use either one. 

That said, some people like the convention of using single quotes for straight constant strings since '$foo' = $foo... while "$foo" = value of $foo.  AFAIK, Drupal coding standards do not specify what can be used for a string.

Adendum:  Note that the string being passed to the t() function should be a static string in order to be translatable.  E.g. if you use "$foo", this would be impossible to translate.  That's what the placeholders are for.