OK,
1. fencing off the culvert - if the goats want to get out they will. I would not use barbed wire on the top. and with pygmies you will need to have tensioned bottom wire. They will go under/though rather than over.
2. Very cautious with badgers - reasons: if your goats kid in the open-fronted shelter the kids may well be taken by the badger, esp pygmy kids - they are tiny. Badgers do carry bTB and goats are susceptible.
3. 4 acres is very big, I would partition, see 4.
4. Do not keep goats and sheep in the same field with shared grazing - they share parasites and goats are much less tolerant of worms and fluke than sheep are. Also ideally goat feed should contain added copper and most sheep do not need it, and indeed can have copper overload pretty quickly.
5. I personally would never leave my goat shed open at night - I also do have badgers around, but also goats really do nto do rain and dampness. So they do prefer a shed that can be´closed on all four sides and is not drafty (as opposed to well ventilated above head height).
I would advise you to speak to other goatkeepers in your area - I am sure the pygmy breeders organmistion have membership lists so you can chat to other goatkeepers and see their set up.
On another personal note, please make sure you get goats that have recently tested negative for CAE , please google it. If you buy in kids make sure the dams have a negative test result in the last 6 months.