There isn't really a difference between losing weight and "burning fat" - burning fat is more of an invented concept. When you lose weight you are typically losing a mixture of water, fat, muscle, glycogen, etc.

Yes, eating fat doesn't necessarily make you fat, it's when you eat too many calories in general that your body will store the excess calories as fat.

There's actually been a lot of recent research that shows that eggs aren't nearly as harmful to our cholesterol as we have been told in the past. This article links to a ton of research, and they conclude that the average person should be able to eat up to 3 eggs per day without any negative results, they also said the following:

Eggs consistently raise HDL (the “good”) cholesterol. For 70% of people, there is no increase in Total or LDL cholesterol. There may be a mild increase in a benign subtype of LDL in some people.

https://www.healthline.com/nutrition/how-many-eggs-should-you-eat#section2

I eat 4 eggs per day regardless of this advice - you could say I live on the edge, but I get my cholesterol levels checked yearly by my doctor and I check out totally fine (I've been eating 4 eggs per day for years now).

Ahh yes, it is interesting that you aren't used to that - I do believe it's more of an American thing to drink tons of coffee. Nearly everyone I know drinks at least 1 cup of coffee per day, it may not be the healthiest thing in the world but coffee is packed with caffeine and "wakes people up" and keeps them "alert" throughout the work day. I drink it for both the taste and the caffeine.

Oil in tea doesn't taste bad actually, it just adds a thin layer of oil to the top of the tea but it has no real flavor.