Check the manufacturer's instructions and recommendations on how to install. They usually specify approved methods and tools for installing. They might also specify the underlayment such as 15# roofing paper. I've heard that strand bamboo is pretty tough and people needing either ultra-strong nailers or needing to pre-drill first. I think a lot of people glue them down for this reason. Your manufacturer might have some approved hints.

If the space below is a crawlspace, make sure there's no moisture issues down there either as moisture can come from below.

For the subfloor, definitely use screws. I would think gluing first is better unless you can think of a reason why you would want to remove the 3/8 layer later on.