All sensible stuff here:
Yes, running high blood sugars can result in weight loss, as the body breaks down fat reserves to get energy. Potentially dangerous side affect is the creation of ketones as a by product.
Importantly, your body doesn't make use of the sugar in the blood, leaving high BS concentrations, which leads to a myriad of complications, as organs try to cope with elevated levels.
Yes, eating less carbohydrate can also result in weight loss as the body breaks down fat reserves to release additional energy. Slightly different from above, when assuming that adequate insulin is present to properly absorb what sugar is there. Of course, the less sugar that makes it into the blood (much of it derived from carbohydrates), the less insulin that you should need - at least, that's the simple theory! In practice, there are so many competing processes going on that the only, surefire way to gauge how much insulin is actually required is to monitor BS.
Fats are complex, as there are many forms, affecting the body in many ways. Having said that, a "low fat" diet can help to reduce the amount of fat that the body stores, helping to reduce weight gain.
The best way to lose weight is by sticking to a nutitionally-balanced diet, matching energy intake to body requirements - easier said than done, of course! 😱
I'd hate anybody to take offence at this simplistic theory, as I have deliberately avoided other complicating factors that beset people in differing ways, making weight control/loss such a complex (and emotive) issue.
The important message is to do what you can to stay healthy!