You should say, "in February." Example: "My birthday is in February." It's better to say only, "this week." Example: "I got a new car this week." Meaning that sometime in the current week, you bought a car. Lastly, you can say, "It happened at work," instead of, "in/on my work." In regards to your last question, you should just say, "sadly." Example: "Sadly, I didn't enjoy the expensive dinner." This rule is true for any adverb. You can normally know that a word is an adverb if it ends with 'ly'.

1. in March BUT on March 1st 2. "this week" alone can work. "During the (last) week" too. 3. It happened at work. (At your workplace.) It happens quite a lot in my work. (In your field, in what you do.)