It is even worse in France: if you code open source "on the side" of you work, at home, the company which employs you may claim the copyrights of it. I had to add explicit exclusion of this claim of copyrights in my job contracts to protect my personal work.
That was a few years back, dunno if that was fixed.
That is not correct; assuming you are not using an employer’s equipment on employer’s time, and/or working on what the employer pays you to do for them or are working on something that is competing and a few other reasonable caveats.
AFAIK it's the same in the USA, that's why one of the first questions when interviewing with a company is to ask them about their moonlighting policy if you do want to work on a side project.