Lightroom is a black box and trying to make comparisons of DT and LR is not going to help you. LR gives you a single slider to brighten shadows and a single slider to darken highlights, but how this is achieved is not really controllable by the user. However. DT’s shadow and highlights module allows the user to control this process if they wish.
Do yourself a favour and download the image and work through the well explained lesson in this link
Also, you are tone mapping, but you are choosing tools not intended to be the primary tone mappers and making them do the job rather than selecting one of the tools designed for tone mapping. The tone mapper is just the start of the editing process and is not intended to give the final look alone.
My view on tone mappers is base curve is old school, it works but there are better options. Filmic is very challenging to learn and master and I no longer teach it to my photography students. Sigmoid gives nice colors out of the box, often without the user having to tweak its controls. AgX is for me magic. I mainly use it by activating it and using the auto tune levels to set the whites and the blacks. I also use the picker for pivot target output. Then sometimes I return at the end of my edit to the pivot target output slider and tweak it a little to ensure the final desired brightness for the image.
Most of your questions can really only be answered by doing some playraw posting with images. Posting your edit here is not informative enough to explain why a tone mapper will benefit your processing.
Good luck with your DT journey and its nice to see you asking questions on the forum.