Nutrient Conversions

The human body is absolutely amazing in the way it's able to build the materials it needs for growth and repair without any hassle.The body can fuel its self on a staggering number of substances.

Can excess protein be converted into fat?

It is almost impossible for protein to get stored as fat. Amino acids are very difficult to turn into fatty acids. The only way for aminos to convert to fatty acids is by converting into glucose first through a process called gluconeogenesis which is very inefficient.

Can carbs get turned into fat?

Carbs can be stored in fat cells in two ways
1 -  Through direct conversion of sugar molecules into fatty acids, this only happens at a slow rate, but even so it can still contribute to 500g or more of pure body fat per year, depending on how much body fat you already possess.
2 -  Sugar conversion into glycerol complexes.
Fat cells don't just suck up fatty acids . They need glue to bind them together into triacylglycerol (TAG) molecules, which are the body's predominant form of stored energy. This requires three fatty acid molecules and one is glycerol.

Fat be converted to glucose?

Yes

Recent studies have shown that certain metabolic pathways exist to allow for a small amount of glucose production from even-chain fatty acids.
Where as odd-chain fatty acids convert easily into glucose. In other words this also means that an ultra low carb diet that's rich in odd-chained-fatty-acids is not ketogenic, and metabolically speaking not much different from a carb based diet.

Conclusion

Not all carbs, fats and proteins act the same in the body. The type you eat makes a massive difference.
Just assuming that any ultra low card diets is  ketogenic is false. Thinking by eating only carbs you can't gain fat is false.

The type, timing and combination with other nutrients is what matters.