Using the Delhi metro from T3 and T1 with luggage

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I would like to use the Delhi metro to get from the international arrivals terminal (T3) to my hotel (yet to be chosen), and then also use the metro to get around to various tourist sites in Delhi, New Delhi, and South Delhi. It seems the Airport Express line, while having a few intermediate station stops, connects solely with the Yellow Line, and that connection is at its terminus in New Delhi. One hotel I am considering is near the Hazrat Nizamuddin stop on the recently opened Pink line. The Delhi Metro "trip planner" listed several options for travel, but all start on the Magenta Line, which stops at the domestic terminal (T1), but not T3. My understanding is that a 5-6 km cab ride is needed to get between the two airports, and that perhaps a flight ticket is needed to even get into the airport (T1), so it seems a bit daunting for someone visiting India for the first time and hoping to just hop on the metro and take off from the airport. Has anyone used the Delhi Airport Express line to get to a metro station beyond the New Delhi station? What is the connection between the Airport Express Line and the Yellow Line like? I presume I would need to pay separately for the airport express and non-express lines. I don't mind riding for a longer period of time or changing trains multiple times if doing so is straightforward. Also, I would be riding very early, probably as soon as the metro starts up at 05:00. I am trying to find a mid-range hotel that I can walk to from my ultimate metro station stop, but maybe that is asking too much. Are cycle- or auto-rickshaws stationed outside all metro stations?