Can H1B and F1 visas co exist at the same time? What is the consular processing?

You can in theory have both a valid H-1B visa and F-1 visa. Whether the U.S. Embassy will allow both will really depend upon your complete immigration history. If you have both, you could switch from one status to another by filing a change of status in the U.S. or entering the U.S. on the other visa. The change of status process takes months so it is usually better to depart the U.S. and return obbthe visa in which status you wish to be. Consular processing is the pierces of applying for a visa at the consult. You can read more at http://myattorneyusa.com/introduction-to-h1b-visas.