Does "allow me" is a more sophisticated form of saying "let me"?

Can I say "I'm very so sorry."?

No, this is an exaggerated form of an apology and the "so" must go before the "very." The only way to use this expression is to say "I'm so very sorry." In other cases where you wish to use "so" and "very" together, the "so" goes first as well. Ex: I'm so very sorry for accidentally hitting your car. Ex: I'm so very sorry for coming late.

What is garden shed?

A "garden shed" is a storage building that holds gardening supplies such as a lawnmower, shovels, a rake, pots, soil, etc. People usually own "garden sheds" if their house is on a piece of land. Ex: Can you get the rake out of the garden shed for me? I need to rake these leaves.

What does "human trafficking" mean?

Human trafficking refers to the illegal act of transporting or forcing people to work or provide some service, usually for the purposes of forced labor, servitude, or sexual exploitation. For example, the act of transporting people to force them to work in a foreign country for little or no pay is called human trafficking. Ex: Human trafficking increases as poverty and economic instability rise. Ex: The trafficking of humans is illegal in every country in the world, but it still occurs on a large scale.

Isn't it right to say "lie down" instead of "lay down" here?

Yes, you are correct. The speaker should have used "lie down" here as she is referring to the act of lying down by a person, which uses the "lie" verb (unless it is past tense, then "lay" should be used.) The two verbs are similar so people sometimes mistake the two. Ex: Lie down on the bed. Ex: Lay the book down on the table.

This sentence seems grammatically wrong. Is it okay?

This sentence is not grammatically correct. To be grammatically correct, it should say "What do you have?" The speaker is speaking in a slang way.