Postgresql – How should I index a UUID in Postgres

It depends on your generation function and size of the final tables

GUIDs are intended to be globally unique identifiers. As discussed in the Postgres 8.3 documentation there are no methodologies that are universally appropriate to generate these identifiers, but postgreSQL does ship with a few more useful candidates.

From the scope of your problem, and the need for offline writes, you've quite neatly boxed out the use of anything but a GUID, and therefore there are no compensatory advantages of other schemes.

From a functional standpoint, the key length is usually not an issue on any kind of modern system, depending on the number of reads and size of the table. As an alternative methodology, offline clients could batch new records without a primary key and simply insert them when reconnecting. As postgreSQL offers the "Serial" datatype, clients will never need to determine the ID if they can perform a simple write to the database.

Assessment

In your last query, the bitmap index scan looking for 'hat' produces 307 hits.
Postgres then runs a bitmap heap scan to filter merchants similar enough ( similarity(...) > 0.2), producing 12 rows. Your test is with 30K rows, so your real life query will produce around 300 times as many hits, 90k / 3.5k for the test case at hand. An additional index on merchant will help.

Advice

I suggest you create an additional trigram index for the similarity search. Be sure to read the chapter in the manual about trigram index support. We need the additional module pg_trgminstalled (like you obviously have).

For your first request:

How can I search for a query like 'WALMART BAGS' which will first return me product BAG with merchant WALMART and then BAGS from other merchants.

I suggest this query using the similarity operator %:

-- SELECT set_limit(0.2)  -- Adjust similarity operator only if needed

SELECT *
FROM   products
WHERE  to_tsvector('english', product) @@ to_tsquery('bag')
AND    merchant % 'walmart'
ORDER  BY merchant <-> 'walmart'
--    LIMIT  n; -- possibly limit to top n results

Again, you can choose between GiST and GIN, but this time GiST carries a decisive advantage:

This can be implemented quite efficiently by GiST indexes, but not by GIN indexes. It will usually beat the first formulation when only a small number of the closest matches is wanted.

Therefore, I suggest this index:

CREATE INDEX prod_merchant_trgm_idx ON products USING gist (merchant gist_trgm_ops);

As for your second request:

Can I have both GIN and GIST index working for me?

Yes, you can. It would hardly make sense to have both types for the same (combination of) column(s), but Postgres can combine GiST and GIN indices very well in the same query. I quote the excellent manual yet again, on Combining Multiple Indexes:

To combine multiple indexes, the system scans each needed index and prepares a bitmap in memory giving the locations of table rows that are reported as matching that index's conditions. The bitmaps are then ANDed and ORed together as needed by the query. Finally, the actual table rows are visited and returned. The table rows are visited in physical order, because that is how the bitmap is laid out; this means that any ordering of the original indexes is lost, and so a separate sort step will be needed if the query has an ORDER BY clause. For this reason, and because each additional index scan adds extra time, the planner will sometimes choose to use a simple index scan even though additional indexes are available that could have been used as well.