(Whether this would stand up in court is dubious, but do you also want to go up against a billion-dollar company's crack legal team in court?)
Also, I've heard the main issue isn't actually getting sued, it's that VCs or investors won't even touch you if there's the slightest hint that you might not own your code. If your side-project takes off it's nice to be able to enlist other peoples' help instead of miring yourself in legalities.
Agree with the VC thing, but assuming you are doing it outside of work it shouldn't be a huge problem, but check with a lawyer.
Why not? These large corporations would pay you $1/hour if the market conditions let them.
The people in companies make the decisions and if you are good to them generally they will be good to you. Don't make bad blood by doing the wrong thing.
Otherwise big companies could claim they own the the capitol improvements to your property which you do on your own time.
Generally don't be a dick and work on your stuff while at work and you will be fine.