I din't know olive oil encourages fat burning either (is it a problem for me?).
No, it isn't :) Olive oil is actually really healthy! As mentioned in the post, it activates sertuin enzyne (which improves longevity) and evidence shows that omega-3 fatty acids which are found in olive oil improve heart health.
There's one Spanish study, where they followed a group of 65-year olds for 4 years, who consumed a liter of olive oil a week (that's like ~ 140ml a day) along with mediterranean diet. The other group was on low-fat diet. The women in the first group had a 68% lower incidence of female-related health concerns.
You never mentioned peanut. Does that have fat or it's more of a protein thing?
They're high in both protein and fat, with a descent amount of fiber. I actually went on to do some research on peanuts this morning. The conclusion is that, you'd want to stay away from peanuts. Studies suggest that the lectin found in peanuts might negatively impact your gut's health. Another compound found in peanuts, called phytic acid can prevent absorption of nutrients your body needs the most. Sadly, sunflower seeds raise similar health concerns, so I'm going to do an evaluation on my diet and on this post.
Fun fact. Did you know peanuts aren't actually nuts, but rather legumes? :)