How soon can I paint treated lumber? And could I used screws or nails on treated lumber?

I don't know how you would put 'treated' lumber together for anything you are building without using screws and nails. You don't have to wait to paint treated lumber.

Paint when dry, Pilot holes if predrilled will prevent your lumber from splitting when nailed or screwed just drill a hole a little smaller then the shank size of your screw or nail before placing nail or screw to prevent unsightly splits and cracks

Don't nail it, the treated wood shinks so much after it drys you nails may not hold like they should, we use 3 screws on the handrails and joists and 2 1/2 on the decking boards, we use nails or nail gun to just get the framing started and that's it, as for paint as long as its not really green and wet , id say a month wait time would be fine, you'll hear of many more months than that but if it looks dry prime it and paint it,