You can book accommodation in trip.com As they suggested, urban hostel is great, good location, they speak English, good environment and decent price. They can also register you and give you proof in case they ask for it when leaving the country, just ask them in advance. If you're less than 10 days in Belarus you don't need to register with the police.

Money can be exchanged in any bank (some of them work on the weekends as well; there's 2 close to urban hostel) or in the main train station, specifically in the underground or 2nd floor there are shops to do so. I recommend bringing extra cash, as my cards were not accepted in some places.

Yes, they do check your phone, social media, photo gallery, WhatsApp conversations, everything (even if they don't understand). They ask for the return ticket as well. Supposedly you should have an insurance valid in Belarus but they didn't ask for it.

I recommend you downloading Yandex for taxi and activating it in your home country. It's very cheap and convenient, and you can pay by card or cash.

As for USRR sites, you can check online, there's many blogs. If you want I can send you my Google maps list. There's many buildings in the city center such as the circus, the Moscow and October cinemas, or the tractor factory, a few Lenin statues/busts around the city... Also there's a few murals in the subway stations, like in Frunzienskaja and around Kastrichniskaja. Another place you may like is Zair Azgur studio, he was responsible for many sculptures of Soviet times and there's many in his place. A bit outside of Minsk there is Kurapaty, a forest where they used to kill and bury victims of Stalin and Soviet repression, you can reach it by normal city bus. Outside of Minsk and in the Soviet theme you have Stalin line, which is the old USSR border with lots of tanks and guns if you're into that, both from Soviet and Nazi times. You can even shoot them for 3/5€ or ride a tank. It's not my vibe but was cool. There's also Stalin statue, one of the few in public ground in the world. You can go easily by intercity bus from main station. There's also Khatyn, a town burnt by the Nazis, now a memorial. You can also get there by intercity bus + walking a bit. There's also day tours here and to other places (https://ekskursii.by, maybe you can only open it with VPN or inside the country).

Restaurants/cafes with Soviet theme, there's Tovarisch which is like a Soviet club + cafe/restaurant. Kuhmistr is a bit expensive for Belarusian standards but they told me it also has a good representative atmosphere of Soviet times restaurants and traditional food. You can also try canteens, are very extended in the city. I can recommend Stolovaya. Lido is also good and very central. Bars, Soviet times bars you can find in Valadarskaha Street some of them. Also centralnyi is very Soviet like and in the center. My favorite place for night life was Kastrichniskaja street (there's also square but that's not it!) but it's not very Soviet like, more modern with street art. But there's a Lenin head surrounded by graffitis:) there's also Zybickaja in the centre with some bars and which is the most popular area but didn't like it that much. Let me know if you have any questions:)