If my offering is on the SINs, do I need to be on the NIST or GSA approved lists?

If you are offering products, the answer is yes. If you are offering integrated solutions, you must be qualified by GSA and you must commit to delivering only products which have been approved and appear on the Approved product List.

Is it required to purchase FIP compliant items only from vendors on GSA's schedule 70 SIN 132-62?

It is required to only purchase products that are on the GSA FIPS 201 APL. However, the method of purchasing this isn't fixed. If the Vendor has the same product on their GSA Schedule 70 SIN 132-62, then it can be purchased from here, or else it would need to be purchased from the open market.

In the event that a vendor offers a product listed on the Approved Products List, but they are not listed under Schedule 70 SIN132-62, what is the procedure that an agency can use to place a GSA order for this product?

Once the vendor is approved from either GSA SIN 132-62 review process or GSA FIPS 201 Evaluation Lab process, they must submit a procurement package to the FAS Schedules program to be considered for an award offer under SIN 132-62. Once the vendor is on Schedule, then any Federal agency or State may procure the service or product. The APL does not mean the vendor automatically receives a Schedule award. If the product is not in SIN 132-62, then agencies can procure the product through any another Schedule that it is on or they will have to procure via Open Market.