Japanese has a wide variety of phrases to relate one sentence to the next. For lack of a better term, I'm going to call these words "Sentence-Initial Conjunctions".
でも, equivalent to "however" or "but", is used when the speaker is giving information that shows a contrast to the previous sentence.
As it is said, no matter how much good things you can tell about someone, if there is a "but", everything goes down to drain. So this "but" is a bit of poison.
In your examples:
①
He has a scary face but a heart of gold.
かれ【kare】は【wa】こわい【kowai】かお【kao】 を{o} している【shiteiru】が【ga】、よい【yoi】こころ【kokoro】を【wo】もっている【motteiru】。
As the tone of contrast from the first part of the sentence completely changes after you say "but" likewise the tone of contrast changes after you say いるが【iruga】, here いる【iru】 means "has", and the particle が【ga】 is attached to the last word of the first part of the sentence as a way of differentiating it from the second part of the sentence.
②
She's very quiet but also very direct.
かのじょ【kanojo】は【wa】とても【totemo】しずか【shizuka】です【desu】が【ga】、また【mata】 ひじょう【hijou】に【ni】ちょくせつてき【chokusetsuteki】です【desu】。
The particle が【ga】 in です【desu】が【ga】 is the contrasting word "but". When you say the sentence, it will naturally carry the same tone when you speak Japanese fluently. It will slightly vary a bit because of the Japanese accent, but the Japanese people will get the tone you are trying to convey.
There is a certain pause of 1 second after part "She's very quiet", then it comes the next part "but also very direct" また【mata】 ひじょう【hijou】に【ni】ちょくせつてき【chokusetsuteki】です【desu】. This is the tone we use in English.
In the case of Japanese, there is a pause after the particle が【ga】 in the clause しずか【shizuka】です【desu】が【ga】.
That is to say, in English the pause comes before the "but", but in Japanese the pause comes after the "が【ga】. Despite the slight variation in tone, Japanese native speakers will get the contrasting tone like the one we have in English.
③
I was doing my best, and yet I also wanted to give up.
わたし【Watashi】は【wa】さいぜん【saizen】を【wo】つくしていた【tsukushiteita】が【ga】、それ【sore】でも【demo】わたし【watashi】は 【wa】あきらめたい【akirametai】と【to】おもった【omotta】。
As you can see here as well, there is a が【ga】 right after the word つくしていた【tsukushiteita】. が【ga】 is used as a connector to give contrast between the two sentences, and naturally, there will always be a pause after it and before the contrasting word それでも【soredemo】("and still"/"but still"), which comes later. This is like the small gap we have after the word "best" (??).
④
I am very happy yet not satisfied.
わたし【watashi】は【wa】とても【totemo】うれしい【ureshii】です【desu】が【ga】、まんぞく【manzoku】 していない【shiteinai】。
It's the same thing here too. The pause is right after が【ga】 in the construction いるが【iruga】 which means "yet" in the sentence.
Note: For most of the contrasting sentences, in Japanese we use が【ga】 for "but", "yet" etc.
I hope this info helped you out. You can try listening to contrasting sentences in Japanese Audio Books online to help you understand better the tone and the pause in the sentences.