The problem with both your constructs is the comma in both the second versions.
- Coming here, I was arrested. The police thought I had some narcotics on me.
- I was arrested coming here. The police thought I had some narcotics on me.
The first one has the emphasis on "coming here", the second on the arrest. You could use the first, for example, as the answer to a question:
"You're late. Did you have any trouble?"
"Yes. First I couldn't find my car keys. I looked everywhere until I found them in the dog kennel. And then coming here..."
The second series is harder, because it's difficult to both read and look at a person at the same time.
A. Reading the third page of the book, he looked at me.
B. He looked at me, reading the third page of the book.
The two versions are more subtle than before. Again the emphasis of the first is on the reading, so him looking at me is probably as a result of what was written there. A more expanded version could be "Midway through reading the third...", saying that he stopped reading to look at me.
The second version is clumsy. At first it looks like it should be
He looked at me reading the third page of the book.
but then it's implying that I'm the one reading the book while he looks at me. Assuming that you want it to be him reading, replace the comma with "while":
He looked at me while reading the third page of the book.
Again, he stops reading to look at me, but this time the reason is less clear. Maybe hes about to say something unrelated?