Any modern day paint even if eaten will not contain lead. All paints when cured are basically inert, as in they are like eating pebbles of stone if chickens/children choose to eat it.
An oil paint takes much longer to dry and cure than an acrylic/latex/water base paint. Acrylic is your most common now a days so it will dry in less than 4 hours and cure exponentially (like concrete) in 28 days. If you've ever freshly painted a kitchen then you know not to wash it for a few days, after 3 days it's a light sponge wash, 7 days and you can scratch fingers or what not and shouldn't leave a mark, 14 days and moving furniture will leave small scuff marks, over 21 days and an impact or scratch will actually dent the substrate, usually sheet rock but in your case plywood. Yeah, 21 days of curing and it's rock solid.