Yes, you can travel out of US, when your H4 Extension is pending. There is no official reference given by USCIS on how they will handle extension...

There may be problems with required continuous presence and possibly physical presence. Consult with a lawyer to review your documents and discuss...

Unemployment insurance payment is not public assistance.

If your visa as principal is revoked, derivative's visa would be revoked as well.

It depends on criminal history. Extensive record of disregard and respect of the law and inability to reform are all grounds for F1 visa denial.

Strong ties to a country may vary. Family and employment background are critical. Additionally, owning a house or apartment, financial prospects...

While 4HED is pending an extension of I94 may be obtained at the POE Mexico or Canada.

No, publishing on research from abroad should not affect your F1 status. In fact, it should be encouraged. Attorney Eddy Chielo, PhD

You do not have an immigration issue. I have referred to lawyers who may help.

If you provide additional information, a qualified immigration attorney may evaluate your case. It's not clear if you have an immigration issue or...