I am an Indian citizen with a valid USA visa. I am flying DEN-ORD-CDG-BOM with United (Denver Chicago) and Delta (Chicago-Mumbai through Paris)

(a) You might have to do security again in ORD, depending on which terminals you arrive and depart from. Probably yes.

(b) You probably won't have to collect your luggage in ORD. There are no exit customs checks in the US. When you check in in DEN, ask the agent if you are unsure where you next need to pick up your bags.

According to What type of visa do I need to transit through an airport in France ?, you should not need a transit visa with a valid US visa (emphasis mine):

Exempted by France from ATV [Airport Transit Visa] are :

• holders of a visa valid for a Member State of the European Union or the European Economic Area, Canada, Japan or the United States ;