Yes, from what I could read in the comments to that video of people who grew up in these Soviet super blocks they had a lot in common. But I am curious what urban planning has turned into in modern Russia? Are any of the ideas of the past still relevant? Is Russia copying urban planning practices from other countries? If so which ones?

Just in case, I want to emphasize that in the USSR, the principles of urban planning were quite different in different periods (and this was said in the video you mentioned). And this, as I understand it, was associated not only with technology, industrial development and expediency, but also with the views of the country’s leadership on the ideal of human life in the near future (but this is a separate conversation).
In modern Russia, there is no unified urban planning concept. It seems to me that chaos reigns in this area. This is reflected in almost all aspects. At the same time, all types of housing are developing: individual houses, low-rise apartment buildings, “town houses”, many-storey apartment buildings are being built. Here are examples from my city (Perm):
https://51.img.avito.st/640x480/8714032951.jpg
https://zvzda.ru/uploads/2759641a332886a443807949913fa257.jpeg
https://lh3.googleusercontent.com/proxy/CpwsuatORazzv2FMKtYfH6aSiLSxAeH_I1goB-D8BnLmM2OyhPtZVh2dTdWWIDAJb1R5cGxmomyBdZ0mRu13y1B7DozIVAmzzBrT6sbizj5QZ4zUejHDNM
https://www.google.com/imgres?

Only the worst experience is borrowed from the Soviet past, the principle of an individual country house is borrowed from foreign experience (but rather limitedly).
Today, it is relatively rare to build entire neighborhoods. Most often, new houses (most often multi-storey ones) are wedged into the existing urban development to free spaces or to the places of demolished dilapidated houses.
Building is carried out according to the following options:
1) “residential complex”. This is the main category of housing construction in modern Russian cities. It is a group of houses in the same style, built by one developer on one project. They are being built both among the already existing buildings and on empty territories on the outskirts of the city. As a rule, the “residential complex” includes only residential buildings, playgrounds and minimal elements of functional and decorative arrangement of courtyards. Parking spaces are located right in the courtyards. If there is an underground parking, a place in it must be bought separately for a very high price (!). Sometimes residential complexes look quite creepy (https://saturn-r-realty.ru/new-buildings/vilgelma-de-gennina-3/vilgelma-de-gennina-3/?utm_source=google_gennina&utm_medium=cpc&utm_campaign=g&utm_content=485577035750&utm_term=%2B%D0%B0%D1%80%D1%81%D0%B5%D0%BD%D0%B0%D0%BB%20%D0%BF%D0%B5%D1%80%D0%BC%D1%8C&callibri=yd_c:{campaign_id}_gb:{gbid}_ad:{ad_id}_ph:{phrase_id}_st:{source_type}_s:{source}_dt:{device_type}). The strangest thing is that so many people are quite satisfied with such housing. But the skeptics gave them the ironic name “cheloveniki”, which can be translated into English as “…”. ;
2) “point building”. This is when one house is being built among the existing houses. Most often, it is multi-storey and does not visually fit well with older buildings. In my city, infill development is formally prohibited, but in fact it is detailed (in this case, one house is assigned the status of a “residential complex” (https://onf.ru/sites/default/files/styles/new_fotog_in_node/public/node_gallery/pushkina_72.jpg?itok=7ZYhg_sj);
3) “cottage village”. This is a suburban neighborhood of individual houses with a minimum functional infrastructure (https://pkk.perm.ru/gallery/ek%20obshiy.jpg). Each house is owned by a resident and has a garden of approximately 600 square meters. The most common materials for these houses are bricks, foam blocks, aerated concrete blocks.

Each flat in a modern multi-storey Russian building has a separate owner. The price of apartments is different, their area and layout are also different. Recently, a huge number of such housing has been built, while in order to ensure an affordable price for apartments, their dimensions and simplicity of planning are increasingly approaching late Soviet standards.

“However, in your article, unfortunately, you did not say anything about the supply of resources to the described neighborhoods. It would be extremely interesting to learn about the organization of their heating, electricity, water supply, sewerage, etc.”
I am not sure if there is all that much to say about it. I think these things are done pretty much like they would have been done in any other western country.

The fact is that there are 2 opposite concepts of resource supply for houses: centralized and individual. This applies to heating, cold water supply, hot water supply, sewerage and even electricity supply. In practice, these are often different combinations of mixed variants. In the late USSR, in multi-storey residential buildings, everything was centralized, in modern Russian individual country houses, as a rule, everything is autonomous, with the exception of electricity and natural gas (although these are often cylinders or gas tanks with liquid propane).
The degree of centralization of each resource is important, since it greatly affects the cost of housing. This is the main justification for the compact construction of multi-storey apartment buildings in Russia.

We definitely experience some heavy Americanization here. In my home town people are driving ever more, and the strip malls and big box stores are growing ever larger. It is not a good development.

The same trend exists in Russia today. Even in the last few years, most of the good little grocery stores in our neighborhood have closed and several chain supermarkets have opened. Rampant motorization leads to the fact that cities are overcrowded with cars (in contrast to the Soviet era, when almost everyone used public transport).
Our liberal urbanbuild oppositionists say we are borrowing the worst Western experience of 50 years ago and that there has long been a trend “in Europe” to prioritize pedestrians, cyclists and public transport. But your words, apparently, indicate that not everything is so simple.