How to select specific row if sqlite database contains multiple rows with same name/features

I'd do it like this:

SELECT
  z.key,
  CASE WHEN z.location is not null 
    THEN z.location
    ELSE o.location
  END as location,
  CASE WHEN z.location is not null 
    THEN z.idx
    ELSE o.idx
  END as idx
FROM
(SELECT * FROM t WHERE idx=0) z
  INNER JOIN
  (SELECT * FROM t WHERE idx=1) o
  ON z.key = o.key

You can use union all with not in

with tab( KEY, LOCATION, IDX ) as
(
 select '01','1-01', 0 union all   
 select '01','1-02', 1 union all   
 select '01','1-03', 2 union all   
 select '02', null , 0 union all   
 select '02','2-02', 1 union all   
 select '02','2-03', 2 union all   
 select '03', null , 0 union all   
 select '03', null , 1 union all   
 select '03', null , 2      
)
select * from tab where location is not null and idx =0 
union all
select * from tab t where idx =1
                    and key not in
        ( select key 
            from tab 
           where location is not null and idx =0
        ) 

 key   location idx
----   -------- ---
 01     1-01     0
 02     2-02     1
 03     NULL     1

Rextester Demo

This is a tricky problem because you are looking at relationships between different rows. This solution is similar to Barbaros's. However, SQLite supports CTEs, so I prefer to express this as:

with l0 as (
      select t.* 
      from t
      where t.location is not null and t.idx = 0
     )
select l0.*
from l0
union all
select t.*
from t
where idx = 1 and
      not exists (select 1 from l0 where l0.key = l1.key);

That is, take everything from l0 -- the ones with idx = 0 that meet your conditions. Then take the idx = 1 for additional keys that do not meet that condition.

First, get only those rows that might be useful:

SELECT *
FROM t
WHERE (idx = 0 AND location IS NOT NULL)
   OR  idx = 1;

key         location    idx       
----------  ----------  ----------
01          1-01        0         
01          1-02        1         
02          2-02        1         
03                      1         

Then use grouping to return only one row per key. If there are two table rows, you want the one with the smaller idx, so use MIN() to select it:

SELECT key, location, MIN(idx) AS idx
FROM t
WHERE (idx = 0 AND location IS NOT NULL)
   OR  idx = 1
GROUP BY key;

key         location    idx       
----------  ----------  ----------
01          1-01        0         
02          2-02        1         
03                      1