Is it necessary for the foreign embassy’s to get registration under GST Law

All UN bodies, Consulate or Embassy of foreign countries and any other class of persons, so notified, would be required to obtain a unique identification number (UIN) from the GST portal. This UIN will be needed for claiming refund of taxes paid by them on the notified supply of goods or services or both received by them.

(ICAI FAQ PUBLICATIONS 06-09-2017 REGISTRATION: FAQ NO. 11)

In case the amount is paid partly to the supplier of service, whether full taxes can be adjusted first? If no, then whether it has to be calculated proportionately?

No, there is no provision under the GST law to allocate part payment of the invoice towards the taxes first so that the input tax credit can be allowed. Second proviso to Section 16(2) of the CGST Act clearly provides that the entire value of supply (with tax) is to be paid within 180 days from the date of issue of invoice. Therefore, as long as the entire payment is made within 180 days, the recipient would be entitled to claim the credit in full.
Assuming that only part payment is made within 180 days, availing of proportionate credit based on such part payment is not provided for under the CGST Act. However, Rule 37 of the CGST Rules, provides for availability of the amount of input tax credit availed of proportionate to such amount paid to the supplier. 

(ICAI FAQ PUBLICATION 06-09-2017 INPUT TAX CREDIT: FAQ NO.7)

Whether the Department through the proper officer, can suo-moto proceed with registration of a Person under the Act

Yes. In terms of sub-section (8) of Section 25, where a person who is liable to be registered under the CGST Act fails to obtain registration, the proper officer may, without prejudice to any action which may be taken under the CGST Act, or under any other law for the time being in force, proceed to register such person in the manner as may be prescribed.

(ICAI FAQ PUBLICATIONS 06-09-2017 REGISTRATION: FAQ NO. 8)

Is possession of a Permanent Account Number (PAN) mandatory for obtaining a Registration

Yes. Every person should have a Permanent Account Number issued under the Income Tax Act, 1961 in order to be eligible for grant of registration under Section 25 of the CGST Act. Provided that a person required to deduct tax under section 51 may have, in lieu of a Permanent Account Number, a Tax Deduction and Collection Account Number issued under the said Act in order to be eligible for grant of registration. However, as per section 25 (7) the CGST Act, PAN is not mandatory for a non-resident taxable person for obtaining registration.

(ICAI FAQ PUBLICATIONS 06-09-2017 REGISTRATION: FAQ NO. 7)

What is the time limit within which the recipient of supply is liable to pay the value of supply with taxes to the supplier of services/goods to avail the input tax credit?

In terms of second proviso to Section 16(2) of the CGST Act, the time limit prescribed is one hundred and eighty days (180 days) from the date of issue of invoice by the supplier of services/goods. If the recipient fails to pay the value of supply (with tax) within 180 days, such input tax credit would be payable by the recipient along with applicable interest.
The above time limit is not applicable to supplies that are liable to tax under reverse charge mechanism.

(ICAI FAQ PUBLICATION 06-09-2017 INPUT TAX CREDIT: FAQ NO.6)