Baby skin. Since babies can absorb through their skin, any oils which are not nut oils are best. Avoid, massage oils based on almond oil for example. Aquaphor ointment is a common oil based moisturizer that can be used.
Baby oil fine... Regular baby oil without perfumes and scents is fine, but it's not necessary for healthy baby skin. What's more important is sunblock and sun protection - especially for those toddlers and babies who are going outside. There are some great "baby" sunblocks which I use on my kids pretty regularly... We usually skip the baby oil.
None if normal skin. Unless baby has skin conditions such as eczema, excessive dryness, heat rash/fungal infections etc...Normal healthy skin of baby is best left alone. Avoid excessive bathing/cleansing of baby's skin. If your baby has concerning skin condition, please consult doc first before treating it. Good luck.