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No, it's generally seen as wise and helpful advice. However, don't use it to a superior unless you have a very close relationship.

It's better not to. It's specifically for 'heavy' tasks. For small things, it sounds overly dramatic.

The most common informal version is 'Bā ye dast do tā hendune nemishe bardāsht'.

Not really. Persian culture emphasizes focus. There isn't a common proverb that encourages doing many things at once.

Yes, it's often used to tell someone they can't be in love with two people at the same time.

Because they are heavy, round, and slippery—the perfect metaphor for a difficult task.

Yes, frequently in editorials, advice columns, and literature.

You can, for emphasis, but 'two' is the standard form.

'Jack of all trades' describes a person's skill level; this proverb describes the *impossibility* of the act itself.

Yes, though they might use the colloquial 'hendune' and 'nemishe' instead of the formal version.