Your father can execute will bequeathing la d to your mother , you and your siblings 

2) on demise of father apply for mutation of property in name of legal heirs 

See splitting khata is not possible father can gift this land to two kids and wife through registered gift deed and the mutation in same khata can be done in favour of two kids and wife. 

Apply for permission to carry on construction on whole land 

You can take approval for that specific part of the land. 

1. Let your father gift the land to his wife and 2 kids, by executing a registered Gift Deed.

2. After execution of Gift Deed, apply to BBMP, alongwith with a copy of gift deed, for change of Khatha from your father's name to his wife and 2 kids' name, based on the execution of Gift Deed. Then the Khatha Certificate can be obtained from BBMP in 3 names (wife & 2 kids).

Dear Sir,

First get a mutual deed of partition executed and registered with the sketched map.
Once that is done apply for Khata transfer. It will be done.

It will be a bifurcation Katha only and not a transfer Katha. Further, if the previous Katha was A Katha, you will get this Katha too as A Katha only.

No split but partition between co owners. Permission will be for the portion on which construction will held.

You can execute internal partition deed for the ownership.  Permission for whole land is necessary

First of all let him partition the property among his two kids and his wife by a registered partition deed.

After that on the basis of this registered partition deed, let them apply for individual khata for their own portion of land acquired through this registered partition deed.

Until and unless the property is partitioned and identified you may not get plan approval for this particular portion of land.

Whether the property has been converted from agriculture to NA?

Yes, you have to take approval from local body.