The physical and mental traumas make my character start games already with damage or horror.
Does that mean that my health and sanity totals are reduced by the traumas? I mean, can't I heal damage/horor normally?
My understanding is that during a scenario I'm able to heal those damages but at the start of each new scenario they would return. Is that right?
You are correct. As Larry says above, Trauma causes you to start with damage (rather than, say, reducing your maximum value) so it could be healed. However, the trauma remains which means you will still start the next chapter with those damage again.