How do “modes of operation” with block ciphers like (eg) DES work? When trying to understand how “modes of operation” are applied when using block ciphers like DES, I noticed I must be missing something. Maybe I simply misunderstood or mixed up things somewhere. 

How are (keeping the DES example) Feistel structure properties achieved, when a mode of operation such as ECB is used?
The block cipher itself is used as a black box for any mode of operation. The internal design - including if the block cipher uses a Feistel network - is completely tranparent to the mode of operation.

The only thing that matters for the mode of operation is the block size of the block cipher. This, together with any usage limitations, limits the applicability of the block cipher for certain modes of operation. (Tripple) DES is less useful for modes of operation such as CTR and unusable for GCM which requires a 128 bit block size.

Note that the block cipher is basically a configuration parameter for the mode of operation. The block cipher itself doesn't use a mode of operation - it is being used by one.