As for your question of the potential for the MDF to off-gas, it will be under the countertop and the fumes will be somewhat contained, but it could still off gas at the underside where it is open to the cabinet below.
It is no longer difficult to find VOC free composite wood products, such as MDF, Plywood or OSB (oriented strand board). As we now understand the health hazards of urea formaldehyde, manufacturers have reduced and/or eliminated them by using Phenol Formladehyde and other non VOC adhesives and binders.
Phenol formaldehyde has similar binding/adhesive properties to urea formaldehyde but does not contain the harmful VOCs. In the past, phenol formaldehyde was used only for exterior grades of composite wood products as it is waterproof. This was typically labeled as marine grade. Now it has replaced the toxic urea formaldehyde.
Look for the NAUF label, which refers to No Added Urea Formaldehydes.
Also, see my answer to the question "I prefer to use MDF for shelving and furniture, but don't want the formaldehyde. Recommendations other than Medite II?" for more information on the potential health hazards of Urea Formaldehydes and some alternate product information.