If you were asked for an original, it should be an original. The signing authority should be able to mail the document within 84 days ... (usual...

They probably can do it. You need to consult with an attorney by showing all the paperwork.

You can find it on your EAD card. Good luck!

What can i do in this situation?

You can enforce your I 864 in a State or Federal court.

What do they they need as a proof of non-immigrant status. H1b approvals I have ... but EAD approvals I don't.

It is impossible to answer your question: ask your attorney. It all depends on specific facts.

I would not advise you to leave the country while you have this discrepancy on the only card you can provide as proof of your status in the US.

Can i travel inside the US with my foreign passport and the i-797?

I would not do that. You might have very hard time coming back.

What should be the correct application type selection in this scenario ?

I might be confused here, but based on the info provided, I believe it should be "change of status."

If I leave the country while my COS to F1 is pending, will it negatively affect my immigration history?

By itself, the fact that you abandoned your COS application should not have negative consequences.

It is possible but I would need to have more information to answer your question.