I've not had any problems with the does getting in or out of the nest box... can't say some doe won't decide the wire nest box is too high for her to jump into. Also I've not used a lid so I can't really say much about that...
Seems to me, just my opinion, that if a doe is going to injure herself getting in or out of any nest box, she would injure her teats. I'd watch for hungry kits who had previously been well fed and if I saw that I'd check to see if the doe's teats looked inflamed and felt hot. Mastitis would make the doe not want to nurse. Again this is something I've (knock wood) have not had in my rabbits, but it does happen and you would need to check with your vet as to what to do for the doe.
I set my nest boxes in the doe's cage instead of installing the box in the floor. My rabbitry set-up won't allow that configuration. Vickie if you have does who drag their boxes around their cage then keep those monsters in OK! Those are no ordinary rabbits!
Raising rabbits is a great experiment in what works for ... YOU and your rabbits. If you and your rabbits are happy, healthy and having fun raising each other then you are doing what you should be doing. If someone, be it two legged or four legged, isn't happy, healthy and having fun then something needs to be addressed.
Go out and pet your bunny... you will feel good, the bun will feel good and the world just might be a tad better... at least your portion of the world. Me, I'm going out and play with my rabbits. I need it.