Design of an application log database
I did the following 

CREATE TABLE L(
Time_Series_TS TIMESTAMP, 
Channel VARCHAR(10), 
Operation VARCHAR(10), 
Function VARCHAR(10), 
Duration INT);


Then

INSERT INTO L VALUES('2014-06-10 09:00:03.457', 'Channel1', 'Operation3', 'Function15', 15);
INSERT INTO L VALUES('2014-06-10 09:00:08.245', 'Channel2', 'Operation5', 'Function10', 22);
INSERT INTO L VALUES('2014-06-10 09:00:22.005', 'Channel1', 'Operation3', 'Function15', 48);
INSERT INTO L VALUES('2014-06-10 09:01:03.457', 'Channel2', 'Operation3', 'Function15', 296);
INSERT INTO L VALUES('2014-06-10 09:01:08.245', 'Channel2', 'Operation5', 'Function10', 225);
INSERT INTO L VALUES('2014-06-10 09:01:22.005', 'Channel1', 'Operation3', 'Function15', 7);
INSERT INTO L VALUES('2014-06-10 09:01:16.245', 'Channel2', 'Operation5', 'Function10', 10);
INSERT INTO L VALUES('2014-06-10 09:01:47.005', 'Channel1', 'Operation3', 'Function15', 20);


I added a few records to your sample for checking. Then ran this query

SELECT MINUTE(Time_Series_TS) AS Minute, Channel, Operation, Function, 
COUNT(*) AS "Count/min", SUM(Duration) AS Duration 
FROM L
GROUP BY Minute, Channel, Operation, Function
ORDER By Minute, Channel, Operation, Function;


Which gave

+--------+----------+------------+------------+-----------+----------+
| Minute | Channel  | Operation  | Function   | Count/min | Duration |
+--------+----------+------------+------------+-----------+----------+
|      0 | Channel1 | Operation3 | Function15 |         2 |       63 |
|      0 | Channel2 | Operation5 | Function10 |         1 |       22 |
|      1 | Channel1 | Operation3 | Function15 |         2 |       27 |
|      1 | Channel2 | Operation3 | Function15 |         1 |      296 |
|      1 | Channel2 | Operation5 | Function10 |         2 |      235 |
+--------+----------+------------+------------+-----------+----------+


Which appears to be the result you want (note 63 as the 1st duration as per my earlier comment). Is this the result you wanted? You can then use HOUR() and DAYOFMONTH() and even YEAR() to aggregate over these also with this query.

For performance, I did create an index 

CREATE INDEX L_Index ON L(Channel, Operation, Function) using BTREE; 


and explained the query before and after creating it, but there was no difference. This is hardly a surprise, since the optimizer probably said that there's no point in using one for such a small table. Obviously, I can't test 
with your data, but there are a couple of points. If you are performing this operation over a large number of records with a large no. of fields, you may run into issues and if you create many indexes, your insert performance will decrease. Is it possible for you to categorise your data in some way to reduce the number of fields - i.e. split your big table into ones with a smaller number of fields? Check out different scenarios, test and see what happens with your data, your queries, your application and your hardware.

[EDIT]

For something more human readable, you might like to try something like

SELECT TIME(FROM_UNIXTIME(UNIX_TIMESTAMP(Time_Series_TS) - MOD(UNIX_TIMESTAMP(Time_Series_TS), 60))) AS Minute,
..
..


for your first field.

[EDIT - Response to UPDATE-1]

OK - so in my schema, you are indexing by (Minute, Channel, Operation, Function)?
See here for the docco on composite indexes in MySQL. If your queries have a predominatly left-right orientation, i.e you [always | usually] query Channel first and then Operation, then Function, you could try an index on Minute + (the usual three). If it's fairly arbitrary, then you could try using 6 indexes, but this will hit insert performance. How much, I can't say, but if this is a DW type app which performs the analysis, you can batch the inserts and only occasionally take the hit for that. You'll have to do a few tests with realistic data and EXPLAIN your queries - with realistic sample data, as I said earlier, the Optimiser with just a few records ignores indexes because the table is too small. Interestingly, on the MySQL man page given above, there's a hashing strategy which looks interesting - take MD5 hashes of CONCAT(Your_Column_List_Here). One other thing that I can suggest is that instead of using the

SELECT TIME(FROM_UNIXTIME(UNIX_TIMESTAMP(Time_Series_TS) - MOD(UNIX_TIMESTAMP(Time_Series_TS), 60))) AS Minute,...


Just remove the TIME() function and then you'll be storing INTs which appears to be better than indexes on DATETIMES - see here for a benchmark. Also as previously mentioned, you should remove your data from Production and perform the  OLAP/DW on another machine. You could also test out the InfiniDB solution that I suggested. It's drop-in compatible with MySQL (no learning curve). Then there are all the NoSQL solutions - we could be here all day :-). Take a look at a few scenarios, evaluate and test and then choose what best fits your budget and requirements. Forgot: Make your OLAP/DW system read only for performing queries - no transactional overhead! Make the OLAP/DW tables MyISAM? This last one is controversial - again, test and see.