Here is how you should write it:
-We are having cold/warm or ( Hot) weather. ( Make sure you put weather at the end because, we do not know what you are having that is cold.)
- At any specific time , At any moment, or you can say, Any moment now. ( This specific moment)
- Correct
- Correct
- Correct
- Correct
- You can say , We are progressing quickly, or We are making progress
- Correct
- Almost all the time, "Yet" is at the end of words when writing, because it is proper grammar, however in your speech it can work in the middle of phrases.
---- Learnt,is not a word we use in American English, but people use it in British English
-- In America we say "I learned"