You can take out all the dead wood now, plus as Sherwin says, up to 1/3
of the live wood. The mulberries weep a lot from cuts, and besides
taking out the dead wood, prune them only in november, so they have
time to close the wound. Mulberries typically need very little pruning.
Do try the berries in june-july, if they are good they are a great
thing to have (no spray, early fruit, shade tree, and all it takes to
get a gallon of them is spread a clean tarp and shake the tree).

If they have not been pruned in decades, it might take you 3 years to
get the apples back in shape. When they start fruiting (which might be
this year, given the stimulation of a hard pruning), prune the immature
apples mercilessly. You should leave no more than a young apple for
every 30 leaves. Obviously, the trees will be Pestville after so many
years so you will have to start spraying right away if you want apples.