No, they do not mean the same.
I have been to London.
Means that you are talking about trips or journeys specifically to London.
I have been in London.
Means that that you are talking about a stay in London (which may be short, or very long, or simply passing through). It doesn't matter whether you took a trip there or not.
"In part" means the same as "partly". If some event was caused by more than one thing, then it was caused, in part, by one of them (note commas). The commas you see there are 'parenthetical' and are used when something is inserted into a phrase. To eliminate that, an alternative is to rewrite the sentences as you have suggested.
Yes, it's a complete sentence. "He has to eat" would itself be a complete sentence, with "more" just a simple adverb to modify how his eating habits should change. A similar example:
She has to study more (if she wants to get better grades).
It's not necessary to specify what she should study. "More" by itself implies the general "more often", or "for ...
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Typically the order in which one puts words changes the emphasis or focus on a particular part of a phrase. Putting any given word first calls more attention it. So, the best way to phrase this depends partly on that. You could even put "Rules" first:
Rules of Association which are Reduced
But that gets so wordy as to ...
You apparently understood the meaning correctly, but there should be no it's in that sentence; and a comma would make it clearer:
Today being a holiday, what are you doing up there?
To grammarians the first phrase is an absolute construction; see explanation and more examples there.
All of these are possible:
I'm reading a book at the same time as listening to music.
At the same time as reading a book, I'm listening to music.
I'm reading a book and listening to music at the same time.
I'm at the same time reading a book and listening to music.
I'm simultaneously reading a book and listening to music.
I have been in London for 3 years (as in the title) means that after spending the said time in London you are still there, while the first sentence (I have been to London) means you have an experience of visiting London (you are not there now).