To answer the question "why does a block cipher use a Mode of Operation", we need to first examine the question "what is a block cipher?"
A block cipher is a keyed operation that converts a string of N bits to a string of N bits (where N is usually fixed by the block cipher; for AES, N=128), in a way that, without the key, looks like a random permutation, but with the key, it can be efficiently computed, both forwards and backwards.
So, with this in mind, why do we use a "mode of operation?" Well, the answer is: when we want to use the block cipher to solve any problem other than converting a string of N bits into another string of N bits.
Some examples of these operations are:
Encrypting a message which might not be precisely N bits long
- Possibly in a way that gives semantic security
- Possibly including a tweak
Creating a message authentication code
Doing both simultaneously (known as a "combined mode of operation")
Generating a random looking bitstream
In light of this, let us look at your examples:
(1) and (2) are obvious requirements we may want from a transform; there certainly do exist modes with do those
(3) might be something we want from a mode of operation; there are modes that can be parallelized as well.
(4) I haven't heard of a mode of operation being designed specifically to do this (if a block cipher is distinguishable from a random permutation, we typically don't use it, rather than trying to design a mode of operation to mask it); however I can't see why a mode can't be used to achieve this.
(5) if one considers 3des as a "mode of operation" of des, well, I suppose we can force-bit these. IMHO, it feels like a bit of a stretch, but I can't think of any technical reason to disqualify it.
So, we see that we might use a mode of operation to do any of the things you listed; however (1) through (3) are far more common in practice.