First, your idea about checking for lead is a wise idea. You can purchase lead test kits at paint stores, hardware stores, or home centers.

Second, I would use a respirator when sanding any lead based paint.(At least use a dust mask)

Third, you can check to see whether the paint is latex or oil by using rubbing alcohol on a rag or paper towel. If the color of the paint comes off on the rag, it is latex.
Otherwise, it is an oil based paint.

Fourth, yes I would prime the surface. I would also recommend the primer be tinted off white. Off white covers deep colors better than white primer.

Fifth, you can use Cabinet Coat, Pratt and Lambert Accolade, or any paint you so desire. Remember that oil paint will yellow over time while latex will hold the true white longer. The only oil paint I like is Pratt and Lambert Red Seal Satin Oil, although they do have a high gloss oil that looks great.

I suggest you use a soft bristle brush. Wooster Series 4169 and 4171 have a soft formulation that will work well with the latex paint products. Oil based paints should use a China Bristle brush for the best results. The Wooster Yachtsman brush works well. On the flat surfaces, I recommend a foam roller. Whizz Pro Foam Rollers work better than the Wooster Foam rollers and comes in 2, 4, and 6 inch sizes. I find they work better than any rollers on Doors and panels.