Hello All, I am a little confused about Dubai Connect.

For: BOM-DXB-IAH.
The fastest trip will be to leave BOM via the 04:30AM flight.
The rest of the BOM-DXB flights later in the day will result in a stop in Dubai and connect to the next day's IAH flight.

I booked the BOM flight leaving at 10AM; consequently will have a stop in DXB.
EK (via chat) said that my itinerary does not qualify for Dubai Connect since I could have taken the 04:30AM BOM-DXB flight and had the shortest trip.

Is this correct?

Another question:
If I stick to my flights and "stop" overnight in Dubai - will I qualify for a chaffeur drive to/from a hotel in DXB or do I need to get a taxi on my own?
(currently there is no option to add a chaffeur drive in DXB on "manage my booking")