Can Postgres use an index-only scan for this query with joined tables?
If additional preconditions for an index-only scan are met, it makes perfect sense to append the column id as trailing column to the index (not as leading column):

CREATE INDEX ON thing_types(first_lvl_type, second_lvl_type, id);


Postgres 11 introduces actual covering indexes with the INCLUDE keyword. 

CREATE INDEX ON thing_types(first_lvl_type, second_lvl_type) INCLUDE (id);


Only a small benefit for your case, but it's a great option to add columns to a UNIQUE or PK index or constraint.

About index-only scans:


The Postgres Wiki
The manual


The most important precondition: The visibility map of table thing_types has to show most or all pages as "visible" to all transactions. I.e. the table is either read-only, or your autovacuum settings are aggressive enough to continuously clean up after writes to the table.

Every additional index adds costs. Mostly to write performance. But also side effects, like exhausted cache capacities. (Multiple queries using the same indexes have a better chance for them to reside in cache.) So it's also a question of size. id is typically a very small column integer or bigint. Makes it a good candidate for the use case.

In particular, adding a column to an index disables the option for H.O.T. updates involving the column. But since id is indexed anyway and typically not updated (being the PK) this is not a problem in this case. Related:


Redundant data in update statements


If you actually get index-only scans out of these indexes most of the time, it typically makes sense to use them. Test with EXPLAIN.

There were limitations for partial indexes in older versions. Quoting the release notes of Postgres 9.6:


  
  Allow use of an index-only scan on a partial index when the index's WHERE clause references columns that are not indexed (Tomas
  Vondra, Kyotaro Horiguchi)
  
  For example, an index defined by CREATE INDEX tidx_partial ON t(b) WHERE a &gt; 0 can now be used for an index-only scan by a query that
  specifies WHERE a &gt; 0 and does not otherwise use a. Previously
  this was disallowed because a is not listed as an index column.