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You can find both product user guide and ATBD at:
https://ladsweb.modaps.eosdis.nasa.gov/missions-and-measurements/science-domain/nighttime-lights/

There is none from LAADS DAAC however you can explore NASA's Worldview portal below to see if there any imagery that helps you answer your question:
https://worldview.earthdata.nasa.gov/

The LAADS post processing tool is the only one available. See:
https://ladsweb.modaps.eosdis.nasa.gov/learn/how-to-use-laads-daac-post-processing-tools/

No, there is no such product available. VNP46Ax are the only Nighttime Lights (NTL) products available. Please also note that NASA Earth science data are not of a sufficient resolution for viewing small object for example emergency vehicles, individual properties, or similarly-sized features.

Dear Marco Laiolo: All LAADS systems are working nominally and haven't received any complain from any other users regarding slow data download. Yes, LAADS data are migrated to cloud but that has nothing to do with users' data download. Also, you need to understand that you are accessing data over pu...

We don't have resources to help you with a particular software/tool such as Matlab. You need to consult with the software user support, user guide, or user forum.

The reason is data outage. No data were collected during June 15, 2001 - July 03, 2001 period because MODIS Instrument on Terra satellite was in Survival/Safe Mode. You can find all data outages at:
https://modaps.modaps.eosdis.nasa.gov/services/production/outages_terra.html

It should be working now.

Yes, LAADS sever is down and our SAs are looking into the issue. We are hoping to have the system restored as soon as possible but there is no estimated time right now.

Unfortunately, we don't have resources to help you with MATLAB related questions. You may want to consult MATLAB code examples provided by HDF-EOS Tools and Information Center at:
http://hdfeos.org/examples/matlab.php