Question4: Q1: Whether a Government Department in Delhi is liable for
(a) deducting TDS on works contract service provided by a contractor (registered in Delhi) on land (in Noida) owned by the Govt Department and the construction is being done at Noida '
(b) deducting TDS on works contract service provided by a contractor (registered in Noida) on land (in Noida) owned by the Govt Department and the construction is being done at Noida '
(c) deducting TDS on works contract service provided by a contractor (registered in Rajasthani) on land (in Noida) owned by the Govt Department and the construction is being done at Noida '

Q 2: Whether the above Government Department in Delhi is liable for registration under GST in all or any of the above situations '

Answer: Section 51 contains provision relating to TDS relevant portion the the section is as follow Section 51: Tax deduction at source (Relevant Rule 12)
(1) Notwithstanding anything to the contrary contained in this Act, the Government may mandate,'' Provided that no deduction shall be made if the location of the supplier and the place of supply is in a State or Union territory which is different from the State or as the case may be, Union territory of registration of the recipient. the tax is not to be deducted when the location of the supplier as well as the place of supply both are out of the state (Delhi in your case ) Now coming to all the situations
(a) Deduct Tax as location of supplier is in Delhi
(b) No location and place both are out side Delhi
(c) Same as above (b)

Answer no 2 In case (a) above registration is required as per Section 24(v) Section 24: Compulsory registration in certain cases. Notwithstanding anything contained in sub-section (1) of section 22, the following categories of persons shall be required to be registered under this Act,'' (i) (ii) ; (iii) ; (iv); (v) (vi) persons who are required to deduct tax under section 51, whether or not separately registered under this Act; Please note that the date from which the tax to be deducted is declared under notification no 33. (Reply dt. 23/11/2017)