Yet is generally used in questions and negative sentences. It does not work in the affirmative sentences which you have posted above. In formal style, yet can be used in affirmative sentences like so:
- We have yet to find out what has happened to him.
- I have yet to speak to the manager about my possible promotion.
You can find more information about te difference between still, already and yet in the Cambridge Online Dictionary.
In your sentences, you should use still or already. Already is used for things that have finished and still is used for things that are still going on at the moment of speaking.
Your sentences would be correct like so:
[ 1] A: We've got to hurry!
B: Calm down. It's still early.
 A: Is he coming back soon? Has he finished his job in Beijing?
B: No, he still has two years.
 We have a long time still. We can go to the gym.
 We are only thirty minutes into the work. But I'm already tired.
In your last sentence, I wouldn't use still or already, but I would use only to contrast the already in the second sentence.