Russia is considered the successor state to the Soviet Union; control of the majority of Soviet state institutions and offices were passed to Russia or dissolved, and Russia inherited the Soviet Union's seat at the UN.

Foreign debts were ultimately inherited by Russia, though it was originally proposed that they'd be split proportionately amongst the former members of the union. A number of the republics wouldn't ratify this and Russia ended up taking on the debts in 1994, with the other former member states relinquishing their claims to Soviet-held foreign assets to Russia in return.

Debts to Western countries were repaid by 2006, however the final debts to former Eastern Bloc countries weren't cleared until 2017, when debts owed to Bosnia & Herzegovina (formerly part of Yugoslavia) were settled.

The registration and use of national flags as trademarks is protected by the Paris Convention. As the flag of the USSR is a former national flag though, rather than a current one, I don't believe this protection applies. There have been applications made to trademark the flag in the EU and whilst these applications have been rejected, these were for reasons other than the protections afforded by the Paris Convention.