It is a common misconception. Knowing two slavic languages from different branches (e.g. west and east slavic groups) improves your comprehension of other slavic langauges a lot, but even that still often not enough. I speak 2 slavic languages but found myself confused with simple everyday phrases in others.

Yup. Grammar is the same just with some different vocabulary (mainly because of loanwords from other languages that got inserted because of history).