There’s a surprising amount of vocabulary difference across Czech and Slovak, in addition to the pronunciation differences you mention. Pre-split everyone on both sides grew up hearing bilingual broadcasts, so they picked up the differences ambiently. I’ve heard that it’s a lot less of a given whether the younger generations presumptively understand each other these days. But I’ve also seen evidence that many still choose to engage with people / content / opportunities on the other side enough to get to solid working familiarity anyway.

This comes from limited first-hand experience and more extensive second-hand cross-generational experience. Take it as you will.

Slovak children usually grow up with Czech narrated cartoons, so they are able to understand Czech more easily. I heard that Czech children does not receive this language training for Slovak, so they have a harder time understanding Slovak language. I never "learned" Czech in school but I watched a lot of cartoons as a child (born '93) and read books in Czech so I have no problem understanding Czech language as a Slovak. I have a hard time understanding Polish though, never clicked for me.

I can confirm this is true. Czech republic is cca 2x the population size of Slovakia and its historically more developed part, so during one state union a lot of media were in czech language and it became our second language without thinking about it. Also Czechs did get a decent exposure to slovakian language.

But if there is no exposure, its becomes visible how grammar is very similar, but most words are just a bit different (very few are completely different), and pronunciation varies so much across whole region (even within given country) that its not easy or even possible to understand each other out of blue, without prior exposure.

I got some exposure to Polish TV during 80s, since commies couldn't put together more than 2-3 channels on TV and those were anyway pretty bland. I can cca understand it, but can't say a single sentence well enough. If I read polish text, I have to read it loud in my head and then I grok it easily, otherwise too much 'cz', 'w', words are too long etc and I lose meaning very quickly.

But in general Polish is a bit further away from either Slovak or Czech languages. We were and still are literal brothers (CZ and SK), extremely similar in so many regards, still see no good rational reason why we split up (of course I know real reasons, but those are nasty as are the people responsible for the split).