How to select info about indexes with specified columns from all databases in the instance using TSQL?
You can do dynamic SQL in a cursor to do it on every database on the server that way : 

DECLARE @DB_Name varchar(100) 
DECLARE @Command nvarchar(1000)
DECLARE database_cursor CURSOR FOR 
SELECT name FROM MASTER.sys.sysdatabases 

OPEN database_cursor

FETCH NEXT FROM database_cursor INTO @DB_Name

WHILE @@FETCH_STATUS = 0 
BEGIN 
 SELECT @Command = 'USE [' +  @DB_Name + '];  select
   idx.[name] as [index_name],
   sc.[name] as [schema_name],
   obj.[name] as [table_name],  
   ips.[avg_fragmentation_in_percent] as [fragmentation_percent]
from sys.indexes as idx
inner join sys.objects as obj on idx.object_id = obj.object_id
inner join sys.schemas as sc  on obj.schema_id = sc.schema_id
cross apply sys.dm_db_index_physical_stats( DB_ID(), idx.object_id, idx.index_id, NULL ,''LIMITED'') AS ips
where idx.[name] is not NULL 
order by [fragmentation_percent] desc;'
 EXEC sp_executesql @Command

 FETCH NEXT FROM database_cursor INTO @DB_Name 
END

CLOSE database_cursor 
DEALLOCATE database_cursor