Yes, moisturize. Babies do need more moisturizing. Their skin does not produce as much natural oil as adults and it is more sensitive. Extra moisturizing is needed because we use all sorts of soaps and cleansers on our babies to keep them clean. All soaps, even gentle baby soaps will remove natural oils and dry out the skin. This is why you should add moisture back to your baby's skin after bathing.
I doubt it. Lotions are heavily advertised to new parents but I have little use for them. Frequent lotion use attracts dust & will sour within a day & lead to more frequent bathing.Bathing can remove natural skin oils, causing dryer skin & more lotion use.I suggest a small amount of petroleum jelly on ankles & wrists that have the most need & alternate bathing in tepid water.
Skin care. Babies have more senistive skin than adults and if you bathe the baby and let them sit in the tub a nd play like most parents do, they can get dry skin. Avoid bubble bath, and please use a lotion with no "scent" because a baby's skin absorbs the chemicals into their system. Try arbonne or another product that is organic.