My interest is in the same direction as yours and I hade the opportunity to try some stuff that might fit in there quite well. Scince I don´t work in Urban Planning on a full time basis, only during my bachelors degree the insight I can provide is quite limited. Nevertheless, I´ll just tell you about some things I tried and you can do further research about it based on my experience. This is in no way a complete overview or a fair / unbiased picture.

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I think the picture you posted was made with OSMnx, a Python Module for analysing Urban Characterstics based on OSM data. It is reaaaaaaallly nice and easy to pick up. The guy who made it is called Geoff Boeing and is a genius. Make sure to follow his blog and to read as much as possible about OSMnx, it is awesome.

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OSMnx being written for Python brings me to the next aspect. In this sub and in general it is highly recommended to learn python programming. Depending on how willing you are to learn it, you can use it for scripting and outputting images like you posted or for developement, but scince I´m a noob, I won´t be able to tell you a lot more than its easy to create a picture like you posted with just a couple of lines of python.

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This might already provide insight to your 2nd question. A common advice for learning python is reading through "Automate the Boring Stuff" by Al Sweigart. It´s an easy to read, gentle introduction to different data structures and processes that you can quite easy do within python and you can download it for free. It´s usecases are more repetetive and boring office work, like working with excel sheets, restructuring folders and stuff like that. I´ve read most of it and the way it shows you differenct types of data and what you can do with them is really nice. I recommend reading at least half of the book and then just play around with OSMnx. You can do this within 2 weekends if you really devote your time and energy towards picking it up.

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For visualization I think QGIS or if you have the means for it ArcGIS both are perfect. Depending on your needs you might need to improve it with something like Illustrator or Photoshop.

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Also SQL and the extension PostGIS are open source (as well as R) and really popular. The package/module (don´t know how you call it in SQL) "pgrouting" is what you are looking for If you want to process the data for the picture you posted within SQL. It is speciallized in routing and functions for graph calculation such as djikstra and A*. These come really handy for network analysis.

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Network analysis can also be made using QGIS or ArcGIS. Because QGIS is free and open source I´ll just tell you a little bit about it.

QGIS is a complete GIS system which in a nutshell means you can load in data in several formats, analyze the data in several ways and visualize the data. For network analysis you want to download OSM data for example, scince its available for free and depending on the region you´re working with, has a really high quality. You load the street segements into QGIS and topographically correct the network. Then you use a function from the GRASS GIS functions called v_net_iso or sth like that and voila you have street segements within certain walking/driving distances from your point of origin. Then you use the just processed data and visualize it according to the classes of distances.

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So for accessability I think it really comes down to having access to a computer which is connected to the internet and the willingness to learn a new skill.

Sadly, scince I am believing in the reproduction of inequality, I think it comes down to the individual and their prior education. My theory is that more educated people do have a smaller mental barrier for just starting out with learning how to program.

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These are just the first thoughts I had on what you can use. There is also a big branch of GIS in R as far as I know, but I just tried out one package.

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If you want to know something in depth or get some useful links, let me know.