Since my application went through the lottery process the first time, can the new company file under cap-exempt? Yes

In some cases, certain strategies can be adopted especially if conditions,plans and circumstances around the RFE have changed

Since a timely filed petition was not filed for you, USCIS is likely to deny the extension and if you are present in the U.S. over 180 days, you...

You can petition for workers, namely, chefs to come to the U.S. and work in non immigrant status. GC can also be applied.

Were you asked to submit any documents to satisfy the rejection? You can apply for H-4 visa as this decision does not prevent you from proceeding...

I hope you are aware that both these might be CIMT and will affect your admissibility in the future. If not, contact an immigration attorney ASAP.

Usually this would have to be explained as wages go up considering inflation, etc and employers are required to pay the prevailing wage

Marriage to USC creates assumption of immigrant intent, this can interfere with non immigrant intent required for entry as F-1 student.

Our office filed H-1B for majority holders and founders under the old rule too and the new rule permits the same with shorter duration of status.

You can apply for F-1 with a pending application, however it will not be approved unless the pending application is also approved