Re: BAX and SBA, I want to say yes but my memory ain't what it used to be and I've shipped upwards of 10,000 packages so it all becomes a blur at some point.
Re: requirements, yes each has their own requirements. Shipsurance, for example, requires that the outer box not show any manufacturer names. In other words, if the box easily gives away what's inside, no coverage. This isn't an unusual requirement, or unreasonable, in my experience. ParcelPro has limitations around certain high-fraud-rate zip codes, which they post on their website. If you ship to an of them, you must make special arrangements listed on their site for those areas.
They get comfortable by laying out these parameters, and holding shippers to them for coverage. I've photographed or taken video of some packaging sessions, to show step by step how the package was created and put together so I have proof that I abided by their requirements and created a sturdy package.