Learn English – Definite article or no definite article before “Americans” in this sentence from “The Adventure of English”

The is appropriate in each of the places you have a question about:

In this experiment, the heat transfer coefficient was calculated, allowing to estimate the frost thickness.

Here, allowing is incorrectly used, since it needs an object (*allowing someone to estimate...*). It would be better to use something else:

In this experiment, the heat transfer coefficient was calculated in order to estimate the frost thickness.

...

Firstly, the air properties are defined.

...

As the crystals grow in radius, the amount of air in the ice is reduced, due to the ice becoming more packed.

As you said, a would not work because in each case, you're talking about a specific pre-introduced or pre-defined entity. There are times when you would want no article, but in these examples, you would want the. See the Wikipedia article:

The definite article in English is the, denoting person(s) or thing(s) already mentioned, under discussion, implied, or familiar. It is often used as the very first part of a noun phrase.

The article "the" is used with singular count nouns (the car) and with singular uncountable nouns (the coffee) and plural nouns (the cars) when both the speaker and hearer would know the identity of the thing or idea already.

However, in English, unlike in some other languages such as French, the definite article is absent before familiar but intangible concepts such as "happiness": Happiness is contagious is correct, whereas The happiness is contagious is not, unless a very specific example of happiness is referred to. The is also omitted when the noun refers to a generic mass object (Coffee grows in Colombia) or to a generic collection of countable objects (Cars have accelerators).

In the first example, Doctor is being used as the name of the person; the doctor is more of a descriptive phrase. It's short for Doctor <his name>.

tend bar is a set phrase, it's a synonym for being a bartender. It's also similar to the way other people describe their work: a mailman could say I deliver mail, a programmer would say I write code, a garbageman would say I collect garbage, and a composer would say I write music. These are all using the noun to refer to the general concept, rather than any specific item, so no article is needed. You would add an article when you need to be specific, e.g. I write the music in TV commercials.