How come the bad roads only seemed to effect the Germans but not the Soviets?
About tanks: When you produce only tanks and weapons while neglecting cloths, food and commodities you end up with a lot of tanks and few commodities. That problem would last long till the end of ussr, when in 70s, 80s there would be a lot of rockets and tanks but you had to stand in 12 hour lines to get wedding rings or 7 hour lines for shoes. Since you can use the population as slaves and anyone who would question your methods would be either shot or send to gulags you have no one who can complain. Very simple really totalitarian economics.
About roads. If you were used to this type of conditions from your childhood you find no difficulty in using them rather then someone who have to adapt. Think of Indians in amazon who lived there all their life and can get through the forests much faster then anyone else. Roads would still be the problem for soviets but they were just more adapted to them + they had their trains for supplies. A genius idea of Tsars who with suggestion from US engineers decided to make different train tracks in Russian Empire because of was cheaper and your enemies would no be able to use them with their trains.
Specifically, what did the Soviet tanks do to navigate their unpaved and muddy roads?
they had roads, when soviets would retreat they would try to blow up roads and bridges, on Kiev road every few kilometers they would put soldiers who would blow up the road if they would see germans. So soviets had intact roads from main cities while german army had to deal with destroyed roads and bridges. Tank can be delievered by train, soviets practiced this type of tank delivery from factories to the crew trainig facilities for example. Later Lend-Lease tanks would be deliverd via railroad the same way.