I'm not sure on the lean forward. My coworker does that but every rep, even with just the bar. It took me a while to get the courage to even mention it. Next time, though, I'm going to film it for her and we'll look it over before trying to figure out how to get her working on the form so she's doing a squat instead of a good morning.

Heels was where the trainer suggested I put my focus, along with making sure my butt is back enough, which sounded funny but it does help me get situated in the beginning. But they also seem to go against the whole letting knees travel much past the toes at the gym I attend for some reason. So, grain of salt with that advice.

If you worry about getting stuck then you may want to get together a plan for what to do in case of failure. Are you using a squat rack or power cage? If you have a place to squat that includes safeties and can't get up, you just let them catch the bar and move out of the way. If you don't have safeties as an option, you can still bail on a lift if you must, it just means a further fall for the weights. So, it depends on your setup how that will go because if it's busy gym and people walking around/behind you then have to be aware of the other factors. At home you may need to make sure the floor can handle the weights too, and what not.

What helped me was making sure I was keeping my chest up, and concentrating on thinking about my whole body coming up out of the bottom, not just pushing with my legs. When I find it happening now, it is often when I stop at the bottom of the squat instead of having a smooth transition from down to up, or when I let my core lose it's tightness.

Your weight should be centred over your mid-foot, so you will be using your whole foot. Now, whether you tip forward because your weight is over your toes, or your weight is over you toes because you tip forward, that is hard to say without seeing you.

I always think about sticking my butt out behind me when I squat. Sometimes when my form flags because I am tired I can tell I am too far forward. But I don't have a concrete "how" or "why" - I can just tell that I didn't put my body out back far enough. Probably just a simple feeling that my balance is totally off.
I always put all my weight on my heels. Occasionally my toes will pop up and I have to readjust because I leaned so heavily to my heels.
Are you low bar or high bar? It makes a difference for your gravity center.

I think someone mentioned the bar should be over mid foot. Leaning or knees going over toes only matters if the bar path doesn't stay over mid foot. Many people get confused over what mid foot means, however, because they think about the part they can see while looking down vs the whole foot.

Here is a handy picture.

Yeah, drop to the floor is pretty much how it sounds and more of a you do it because you have to situation. That's why options like the cage or rack are better safety wise and less damaging to equipment/floor/others/you. It's more of a last minute, desperate, uh-oh kind of fail. Not recommended but exists. Just like with bench, roll of shame or the "oh $#!!" rack spot are probably better than tilt and let plates drop but all plausible for getting out from the bar when you can't push it back up.

I use sneakers too, though not running specific. I have a really old pair and a pair my mom gave me which I use both for work where I'm on my feet all shift. Now that I'm not doing cardio after lifting, I could probably figure out a different shoe for lifting but I haven't yet as that involves money. Maybe some day I will get flat shoes and micro plates. *dreams*

I actually do high bar. It's what I was shown when I started at the gym and when I asked the guy during a form workshop he didn't seem inclined to want to work on different forms or changing anything up. I hold it low bar for good mornings but for squats except with really light weight when I'm trying to do it on purpose, I do high bar.