Non-Clustered Index key columns to cover a variable where clause
That's a great question.
And there are good answers.  

The engine definitely will use the index even if you don't use every key column.
That's especially so if they are in order, as you are talking about.
(can anyone else speak to different orders of key columns?)

You will benefit just fine from selecting just on the first column alone as a key, or multiple columns.
What will make a difference - for any index - is staying inside the INCLUDEd columns.
No matter how many key columns you use in your Where, the performance hit for having to go back to the primary key for additional columns can be huge as it doubles the "operations".

When it comes to dealing with performance vs. size, you have the same problem as with any index.
Since you know you want the same columns returned in all cases, if you are READ focused, you will probably want to the index with all 6, if you INCLUDE everything.
It will certainly save you db size compared to making both indexes.

On WRITE, you obviously have a bigger burden with a larger index.  That is a significant additional amount of sorting.
If you do just one row inserted at a time, maybe it won't hardly matter at all.
If you do bulk inserts, you'll definitely want to test the two indexes to see the write performance for your actual inserts.