That depends the hospital. I was not given depends (which are also disposable underwear) I was given mesh disposable underwear with large medical pads inserted in the underwear and put under me. If you are concerned about what they will give you, you will have to call the hospital not all of them are the same.
The disposal panties r depends underwear. After u deliver the hospital gives u some for the bleeding since ull be bleeding heavy. N will be sleeping to recover. It's easier then pad (which they will give u also) but u can just throw them instead of throwing ur regular undies. I got some depends for me for my hospital stay. Cuz sometimes the ones the hospital gives u r too big. But it's mostly for that it's convenient.
Yes mesh panties are the disposable panties you do put a pad in them. I also recommend bringing your own pads to the hospital. The ones they gave were awful. Trash bag and towels would prevent your water damage to the car, but unless you've been told otherwise it's not that common for your water to break all at once.
And I think disposable panties might be mesh ones the hospital provides for you ( can't be 100% sure I'm Aussie not American and we don't get them here) but yeah from what I understand you put the pads in them and toss the whole thing out when your done
Garbage bag over the seat and old towels on top of the garbage bag.