What is the ruling on specifying the Month of Rajab for Umrah, or Fasting or any other righteous actions? And does it have a distinction over the rest of the Sacred Months?

There is no distinction for the Month of Rajab over the rest of the Sacred Months, and it is not to be specified:

Not for umrah, nor fasting , nor prayer and it with reciting of the Quraan, rather it is like the rest of the Sacred Months.

And all the reported narrations regarding the virtue of prayer in it (Rajab), or fasting in it, indeed they are weak, and legislated ruling is not founded on it.

Al-Liqaa’ Ash-Shahri 14/32