No. this will not cover the two indexes.
Lets review those:
IDX_1 = (COL1), (COL2) DESC
This one is indexing by COL1 as primary and COL2 as secondary
IDX_2 = (COL1) INCLUDE (COL3)
This one is indexing by COL1 one exclusively. the INCLUDE is just a data column added to the index, not indexed by.
IDX_3 = (COL1) INCLUDE (COL2), (COL3)
This index is ordered by COL1, just like IDX_2, include in its data, the information from the other two columns.
Depends on your queries you could have try index like this:
IDX_4 = (COL1), (COL2) INCLUDE (COL3)
This is ordered by COL1 First, Secondary is COL2, Including Col3 data in the index but not relevant to the indexing order.