And are we eligible to remove restrictions while applying for Removal of Conditions?

This is the last question, and I will shut up then.

Once you have a green card you are eligible to file for an unrestricted SSN. If, however, your green card is expired while you are waiting for the Removal of Conditions I think you will have trouble. I am not sure what SSN will do if your greencard is close to expiring -whether they will say to wait for your 10 year card or what. Even though the extension letter does extend the greencard for a year, the SSN office doesn't seem to understand it as sometimes they will grant the unrestricted SSN and at other times they don't, so it is inconsistent. The best idea is that when you get your first greencard you apply then for your unrestricted SSN card. There are no fees involved with getting or changing or updating a SSN.

NO FEE.

You need to take care of the SS card soon after getting a green-card, SSA wont deal with you after the card expires, so do it NOW if your green-card is still valid.

Can we do this even if it is a conditional Green Card?

Documents that establish lawfully admitted for permanent residence (LAPR)

-- I-551 card

NOTE: Some LAPR aliens have conditional 2-year I-551s (Alien Registration Receipt Card), subject to the limitation that they must apply for removal of the conditional basis 90 days before the second anniversary of the admittance date to the United States. A conditional I-551 is identified by an expiration date 2 years later than the admittance/adjudication date is acceptable proof of permanent residency.

-- Temporary I-551

Temporary evidence of immigrant status is issued for an alien to use until the permanent I-551 card is received. This may be a stamp in the individual's passport or on the I-94.

RM 00203.410 Evidence of Alien Status for an SSN Card for an Alien Lawfully Admitted for Permanent Residence:

http://policy.ssa.gov/poms.nsf/lnx/0100203410

B. GLOSSARY

10. Conditional Resident Alien

A conditional resident alien is an alien granted a 2-year period of permanent resident status based on (1) a “qualifying” marriage to a U.S. citizen or national, a permanent resident or (2) qualifying as an alien entrepreneur. A conditional resident alien has the same INS documents as an LAPR except that the I-551 expires after 2 years. For enumeration purposes conditional resident aliens are treated as aliens lawfully admitted for permanent residence.

RM 00203.400 Who Must Submit Evidence of Lawful Alien Status for an SSN Card:

https://secure.ssa.gov/apps10/poms.nsf/lnx/0100203400