First check the reason for the denial. Perhaps she violated her status. Likely she can consular process for the visa.

They cannot change to H4's then. They can try consular processing or return to their country for two years.

You mean your sister filed a visa petition for you? It should not prevent you from winning the diversity visa lottery.

You are not required to update this information with USCIS.

No, there should be no problems to be admitted to the U.S.

Get and keep about twenty certified dispositions, since they will ask you every time you return to the U.S.

You could try. It may or may not make it into the file or come to the attention of the examiner.

Not recommended. Yes, CBP could and likely would question your qualification for admission as an E2 if you have an immigrant visa pending.

They will detain you and schedule you for a "credible fear" interview to decide whether you can pursue your application in the U.S., or whether to...

If you have had no encounter with DHS, then you have not been ordered deported.