safety: Probably not the best thing to play in, but definitely not the worst. I wouldn't let my kids play in/around it if I could prevent it.
Why are you going to all the bother to remove and replace? If it isn't moldy or mildewy or wet then it should be ok to use again. Any moisture or water issues aren't going to go away just by replacing it with new, fyi. If you need to remove it then just get some old clothes on, long-sleeve, etc., wear a hat and gloves and goggles, etc. try to eliminate any exposed skin as it can get itchy. Wear a decent 3M dust mask, at least, and keep your shirt untucked so the dust won't creep into your pants - ITCH! You might buy one of those relatively cheap tyvek coverall disposable worksuits that painters, etc. use instead. I would buy the largest sized leaf bags to put it in. If you roll up the batts they should compress down pretty far, maybe duct tape them so they stay small - easier to dispose of. When you are done you will still be itchy, no matter what. Run your itchy clothes through the washer and dryer alone - don't mix with other household laundry or those might end up a little itchy. Wash up and then shower, at first use ONLY cold water (or mostly cold) and don't switch to warm/hot until you have scrubbed yourself twice. If you immediately use hot water then your pores open and the itch goes right in and you will suffer. If you are concerned about dust going into the rest of the house then close all basement doors and windows when you are doing it, except have a fan blowing out of one, cardboard and tape around it for a good seal and all the dust will go right out. Maybe open a window upstairs (other side of house from fan) to prevent negative pressure, or just turn off your furnace. When you are completely done then you could vacuum walls/floor, etc. if you want to remove all dust.