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'.