Learn English – This book is the property of

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.

When the value is subject to change, then has a value of N is correct, it implies at different time or circumstances, the value may be different. Over time, the train will move at lots of different speeds. The value of gold changes depending on the market. The value read from a capacitor changes depending on charge time and voltage.

He has a major in social work. This implies he could have other majors in different subjects.

It has the color of rich deep caramel. This implies the color property does not change. Whatever "it" is, we wouldn't expect it to be blue tomorrow or black yesterday.

Using a or the implies variability in the value. There will be many examples, as you have cited above, that appear inconsistent with this. I'd put it down to most writers not being philosophers or logicians, and they write what sounds appropriate, using a or the almost interchangeably.