Can bad table-scan queries slow down nice index-covered queries at the same instance?
I can definitely say "Yes, your bad query can impact Ivan's query" due to resource competition at instance level.

I have multiple databases in one production sql server box, I have seen many times that a bad query (actually a SP) in one database that caused both CPU and I/O usage skyrocketed and impact all other user sessions connected to other databases.

Besides CPU and I/O usage, one particular case is the negative impact a bad query brings to TempDB performance, and when TempDB is heavily utilized, it is a direct hit to all other queries that need to use TempDB for their executions.