PostgreSQL Index Caching
Playing a bit with pg_buffercache, I could get answers to some of your questions.


This is quite obvious, but the results for (5) also show that answer is YES
I am yet to set up a good example for this, for now it is more yes than no :) (See my edit below, the answer is NO.)
Since the planner is who decides whether to use an index or not, we can say YES, it decides caching (but this is more complicated)
The exact details of caching could be derived from the source code, I couldn't find too much on this topic, except this one (see the author's answer, too).  However, I'm pretty sure that this again is far more complicated than a simple yes or no.  (Again, from my edit you can get some idea - since the cache size is limited, those 'sensible' indexes compete for available space.  If they are too many, they will kick each other from cache - so the answer is rather NO.)
As a simple query with pg_buffercache shows, the answer is a definitive YES.  It is worth to note that temporary table data don't get cached here.


EDIT

I've found Jeremiah Peschka's terrific article about table and index storage.  With information from there, I could answer (2) as well.  I set up a small test, so you can check these yourself.



-- we will need two extensions
CREATE EXTENSION pg_buffercache;
CREATE EXTENSION pageinspect;


-- a very simple test table
CREATE TABLE index_cache_test (
      id serial
    , blah text
);


-- I am a bit megalomaniac here, but I will use this for other purposes as well
INSERT INTO index_cache_test
SELECT i, i::text || 'a'
FROM generate_series(1, 1000000) a(i);


-- let's create the index to be cached
CREATE INDEX idx_cache_test ON index_cache_test (id);


-- now we can have a look at what is cached
SELECT c.relname,count(*) AS buffers
FROM 
    pg_class c 
    INNER JOIN pg_buffercache b ON b.relfilenode = c.relfilenode 
    INNER JOIN pg_database d ON (b.reldatabase = d.oid AND d.datname = current_database())
GROUP BY c.relname
ORDER BY 2 DESC LIMIT 10;

             relname              | buffers
----------------------------------+---------
 index_cache_test                 |    2747
 pg_statistic_relid_att_inh_index |       4
 pg_operator_oprname_l_r_n_index  |       4
... (others are all pg_something, which are not interesting now)

-- this shows that the whole table is cached and our index is not in use yet

-- now we can check which row is where in our index
-- in the ctid column, the first number shows the page, so 
-- all rows starting with the same number are stored in the same page
SELECT * FROM bt_page_items('idx_cache_test', 1);

 itemoffset |  ctid   | itemlen | nulls | vars |          data
------------+---------+---------+-------+------+-------------------------
          1 | (1,164) |      16 | f     | f    | 6f 01 00 00 00 00 00 00
          2 | (0,1)   |      16 | f     | f    | 01 00 00 00 00 00 00 00
          3 | (0,2)   |      16 | f     | f    | 02 00 00 00 00 00 00 00
          4 | (0,3)   |      16 | f     | f    | 03 00 00 00 00 00 00 00
          5 | (0,4)   |      16 | f     | f    | 04 00 00 00 00 00 00 00
          6 | (0,5)   |      16 | f     | f    | 05 00 00 00 00 00 00 00
...
         64 | (0,63)  |      16 | f     | f    | 3f 00 00 00 00 00 00 00
         65 | (0,64)  |      16 | f     | f    | 40 00 00 00 00 00 00 00

-- with the information obtained, we can write a query which is supposed to
-- touch only a single page of the index
EXPLAIN (ANALYZE, BUFFERS) 
    SELECT id 
    FROM index_cache_test 
    WHERE id BETWEEN 10 AND 20 ORDER BY id
;

 Index Scan using idx_test_cache on index_cache_test  (cost=0.00..8.54 rows=9 width=4) (actual time=0.031..0.042 rows=11 loops=1)
   Index Cond: ((id &gt;= 10) AND (id &lt;= 20))
   Buffers: shared hit=4
 Total runtime: 0.094 ms
(4 rows)

-- let's have a look at the cache again (the query remains the same as above)
             relname              | buffers
----------------------------------+---------
 index_cache_test                 |    2747
 idx_test_cache                   |       4
...

-- and compare it to a bigger index scan:
EXPLAIN (ANALYZE, BUFFERS) 
SELECT id 
    FROM index_cache_test 
    WHERE id &lt;= 20000 ORDER BY id
;


 Index Scan using idx_test_cache on index_cache_test  (cost=0.00..666.43 rows=19490 width=4) (actual time=0.072..19.921 rows=20000 loops=1)
   Index Cond: (id &lt;= 20000)
   Buffers: shared hit=4 read=162
 Total runtime: 24.967 ms
(4 rows)

-- this already shows that something was in the cache and further pages were read from disk
-- but to be sure, a final glance at cache contents:

             relname              | buffers
----------------------------------+---------
 index_cache_test                 |    2691
 idx_test_cache                   |      58

-- note that some of the table pages are disappeared
-- but, more importantly, a bigger part of our index is now cached


All in all, this shows that indexes and tables can be cached page by page, therefore the answer for (2) is NO.

And a final one to illustrate temporary tables being non-cached here:

CREATE TEMPORARY TABLE tmp_cache_test AS 
SELECT * FROM index_cache_test ORDER BY id FETCH FIRST 20000 ROWS ONLY;

EXPLAIN (ANALYZE, BUFFERS) SELECT id FROM tmp_cache_test ORDER BY id;

-- checking the buffer cache now shows no sign of the temp table