I am not so sure about that expanding foam. Generally one uses roofing tar on a leak. But before patching, make sure your flashing is correct at your roof joint. You may even wish to hire a roofer or carpenter to make the repair - it has to be water tight and hold up to the sun and the elements for years.
With respect to the lathe and plaster, it almost looks like you would be better off stripping all of this down to the studs and putting up drywall. But first make sure the leak is fixed! For drywall finishing info, go to
Taping Drywall Step by Step Instructions - How To Tape Drywall
Finishing a stair area is always difficult. You may wish to build some temporary scaffolding.