Fortunately (or unfortunately?) I don't work for one of those. I just try to make sure I don't use any company resources (time, equipment, etc) when I work on my own stuff and I work on stuff that has little/no overlap with any of my employer's businesses.
If you have questions, you should review the documents you signed when you started working for your employer, and clear it with your management. Yes, talk to a lawyer if you must, but that would be due to an abundance of caution.
A) You signed a contract that explicitly assigns ownership of anything you create to them B) You created it on equipment they own, or used any of their resources to make it C) You created it in time you should've been working for them
Generally speaking, the best way to avoid legal problems is not to piss people off too much. So don't build a direct competitor to them and you're probably fine. If you haven't done any of A, B or C, you'd probably win eventually, but could still end up spending money on lawyers just to make it go away.
Probably not. If you are doing it on your time and not using company resources any claim to the outside work by your employer are going to be pretty weak.
If you are building a product that directly relates to what the company does you are probably entering a gray area.
If it's worth it to them, they may try to claim it anyway. Depending on applicable laws. Or how far they think they can strongarm you. </IANAL>
That said, the basic gist of it something roughly like this: Some companies may ask you to sign an agreement when you accept employment, which assigns them rights to anything you create, no matter when or where you create it, or what the subject matter is. For example, when I worked for Lulu.com, the initial IP assignment they asked me to sign was so broad, that if I had written a romance novel in my spare time, they would have had a claim to it.
However, some companies have contracts that are much more reasonable, specifying that they only own stuff you create that is directly related to your line of work, and explicitly excluding stuff you do after hours, on your own equipment, not related to your job there.
Other companies may not ask you to sign any such agreement at all.
So, part 1 of answering this question involves finding out what exactly you signed, if anything.
But... it gets more complicated. State laws may trump your IP agreement with your employer. Some states don't allow employers to make those overly broad "we own your brain 24x7" type claims, so even IF you signed such an agreement, it isn't enforceable.
See what I mean about about talking to a real lawyer (preferably one who is a member of the bar in your local area)?
Another angle is this: Most companies aren't actually flat out evil and don't want to screw you over because they're dicks. I know, this totally violates the left-wing narrative you may have been led to believe, but trust me. Anyway, if they DO have a "we 0wnz your brain" agreement, it is entirely possible that some overzealous lawyer (who may have done it on contract and not even be an employee) wrote the thing (or they downloaded it from some form-paperwork website) and the company barely even knows what is there, and wouldn't actually ask to 0wnz your brain if they thought about it. In those cases, if you simply ask them to change the agreement, there is a good chance they will change it, or just ask you to submit a letter listing the exceptions you want to make, and which they can then sign. Again, by way of example, when I was at Lulu, this is the way they handled it when I pushed back. They just said "OK, just give us a list of everything you are working on on your own time and as long as it isn't directly competitive with us, we'll exempt that". I did, and that was that.
So, it can be complicated, but it isn't always a big deal. The other factor to consider is that if your startup has like NOTHING to do with what your employer does, it's entirely possible that you can start on it, work on it, later quit when the startup gains traction, and nobody will ever be the wiser. But it is a little bit of a risk. If you go that route, I'd advise you to never, ever, ever mention the startup to anybody at your other job and keep a pretty low profile about it. But really, I think you'd be better off spending a few bucks on a laywer (if necessary) and / or do some negotiating with your current employer and get everything squared away nice and proper first. Why take on unnecessary risk?