general use of the preposition "with"

[LouieL481]

Dear Ron and Tim,
I ran into a question earlier regarding the use of "with". To avoid any copyright issues, the re-worded question is the following:
The structure of the eye, _________ , helps explain.....
The answer is : "with its miniature eyes called blabla", not "having its miniature eyes"

If I am not mistaken, the +ing form modifies the whole phrase and "the structure having eyes " is wrong since the correct subject predicate having is the eye not the structure.
As for ",+with", does it only limit to the noun before the comma or could modify the whole phrase as well?

I randomly made up an example:
The man in black, with his hands full of popcorn, is heading down the hallway.
In this sentence(if grammatically correct), "with" should describe the whole phrase before the comma.

Could you please briefly summarize the use of "with" if possible?
Your help is much appreciated!

[RonPurewal]

if the blank contains "with xxxx", in which xxxx is a structural element of the eye, then this difference is a non-issue.

in that case both interpretations are valid:
• xxxx is a component of the eye,
• xxxx is a component of the structure of the eye.

so, effectively no difference.