凭, 按, 根据, 依据, 依照, and 按照 how to use these words correctly?
There are actually two ways to answer your question. I'm not sure which would be more helpful to you:

1) an etymological answer: why are these particular words being used in this particular way;

2) a grammatical answer: what syntactic differences exist between the words; what different kinds of sentences are they used in.

For me, I often find the etymological answer is more useful, in that I get more of a sense of what the speaker is thinking. Grammar often does not help that much in illuminating what someone is thinking, though it is essential for using the word correctly yourself.

I'll try the etymological approach here, and look at what 按, 照, 依, 憑, 根, and 據 are doing here.

按 = to press with the finger; imagine the judge running a finger down a list of rules.
依 = to lean on; to rely on something to hold you in a set position.
照 = to illuminate; in fact, this is a key word: you could take it to mean 'according to' all by itself, with modifications indicated by what's in front: 依照,  按照 are variations of 照; they both rely on light to some degree

憑 seems to be 'rely on' in a mental sense; it is closer to 靠 in meaning than 依. Of all the words here, 憑 is the furthest removed from the central meaning of 'according to'.

根 means root; in a more fig. or adverbial sense, it means to take something as a base, what something is built on
據 is another key word; it originally meant to support oneself with a staff. Light is not necessary for this. If one's staff is touching the ground (rooted, 根據) one knows one has a solid basis for action. One can also lean on one's staff 依據, and avoid slipping.

So basically there are two ideas here: 
照 is to follow something that illuminates one's way.  This is often realized as following someone's rules, methods, or example.
據 is to have a fixed point that allows one to act without mishap. This often suggests the idea of a basis for one's actions or beliefs

These two basic ideas can be modified by appropriate forms. 依 works with both 照 and 據; on the other hand 按 doesn't work with 據 (why press on a staff with one finger?) and 根 doesn't work with 照 (what is the basis of a lamp?)

This is what I see as the semantics involved; for grammar, the comment under your answer has perhaps more useful guidance.