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Well, this one is pretty foggy, which is why I missed the first time around and then had to break open my grammar book.

I think the bottom line is that the phrase does not function as the subject,...

By your desription of the phrase, it cannot be a gerund. Gerunds function as nouns, they do not describe things.

If its a noun, what is its function in the sentence?

Here is the sentence restructured to make the phrase a gerund.

Illustrating with RAO EES and Sibneft’s examples helped President Putin to...

I'm certain it is a participial phrase. First, as was pointed out, it doesn't function as a noun (I failed to run it through a gerund check before I answered, and I apologize to the original...

You are correct. It is not a gerund but part of a participial phrase.

Illustrating with RAO EES and Sibneft’s examples, President Putin has recently said that Russian companies pay too large of dividends.

I believe the first word is a gerund.