No, you don't have to be on H-4 to be listed on the forms.
But, you are wise to be concerned about immigrant intent .. .I recommend that you don't travel out of the US until the process is complete.
I'm not sure what type of cross-changeability you're thinking about.
Talk to the company's immigration lawyer, not a non-attorney in HR.
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For cross chargeabilty purposes what nonimmigrant category you happen to be at the time does not matter at all.
Your husband can add you on his I-40 petition while you are on F-1 status. It is perfectly OK.
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