Brand new means that you are the first person to have owned it. You use to emphasize that you are the first owner because we can say new as in new to you. For example, I bought a new car can mean new to you or it can mean that you are the first owner. I bought a brand new car means your are the first owner Andres: I bought a new car. Your friend: Is it brand new? Andres: Yes! (means you are the first owner). Andres: I bought a new car. Your friend: Is it brand new? Andres: No. (means you bought it used but it is new to you). Of course some things are implied to be brand new, like toothbrush, food, etc. Just new is not really used much.

An ufc fighter, after defeating his opponent said: “I said I’d punish him”. I assume the fighter said that because he had a feeling of what -would had happened, right?- The UFC fighter probably said it before the fight and then reiterated and emphasized on what he said was going to do. In this case, you would have assumed the fighter said he would "punish him" his opponent even though you never heard him say this. I never thought Trump would win - Here the conditional part is that he won. If he hadn't won then your phrase wouldn't have made sense.

You mean I am on a website. If so, then number 2.

It's a complete sentence but it just wouldn't make sense if you just said "Air is to us".

Rope is like string but much thicker. You can pull things with rope and you can climb things with rope. The sentences you are hearing are metaphors. Holding on the rope is a metaphor for holding onto to something like hope, or keeping someone (holding onto them). Lost the rope is a metaphor for you lost something.

PTE ESSAY

Urban sprawl has become a controversial issue nowadays. Although some people believe that vertical cities are a remedy for the problem, others support that architects should stick to the traditional way of construction. In my opinion, vertical rather than horizontal cities do not resolve the problem whereas many other issues may occur. This essay will discuss both sides of the argument, providing a reason why people should not alternate the traditional forms of cities.
Nowadays, internal and external immigration has led to overcrowding. In some respects, that issue could be resolved whether the construction of cities change to vertical. According to different types of research, under particular circumstances, vertical buildings could offer ample space, resolving gradually this major problem. Therefore, vertical buildings might constitute an immediate solution for overcrowding.
On the other hand, changing the basic policy, and formation, of the modern architecture, could provoke significant problems rather than resolve one. The main argument is that physics has certain rules such as gravity and, thus, any experimentation will eventually lead to collapses. For example, in case buildings double their capacity, that possibly would create issues regarding the stability of the construction. Therefore, it has been proven that a potential modification in the conception of building's construction entails important implications in its consistency.
In conclusion, urban sprawl is a debatable issue which emerges. As this essay has argued, any change in the architecture of buildings would lead to unpredictable devastation.

I only see one mistake.