Learn English – Difference between a hole and a pit

Virtually means "almost (in respect to degree or strength)" or "almost completely".

In your first pair, the difference is very small:

This mission is almost impossible to achieve.

This mission is virtually impossible to achieve.

Here, I think virtually is a stronger choice, so the second sentence simply means the mission is more difficult than the first. Both sentences mean basically the same thing.

Your second pair has a much bigger difference:

He was almost killed in that accident.

He was virtually killed in that accident.

Here, the first sentence can mean either:

• He was almost hurt in an accident, but he wasn't hurt at all. (For example, his car almost fell off a cliff, but it didn't fall. He got out safely.)

• He was hurt very badly in an accident. He almost died from his injuries.

On the other hand, the second sentence (with virtually) can mean only the second sense. He was hurt badly and was close to dying.

Perhaps a good rule is to imagine that the adjective modified by virtually really is true, but then take a tiny step back:

He was killed in that accident. He's dead.

He was virtually killed in that accident. He's been in the hospital for a month.

"To inaugurate" is fairly archaic in American English. We say people get inaugurated in political office, or to inaugurate a new policy. I couldn't tell you exactly what the difference is, but I can't remember the last time I heard someone use the word for anything but a new president.

To say something "launched" implies movement, and a certain element of risk. Its usage is metaphoric, as literally speaking only vessels launch. When you "launch" a new software, you are taking it from its safe harbor (the development team) and putting it out into the dangerous seas (the general public). It used to imply adventure and risk, but is now sufficiently commonplace that people don't normally think of it that way. "Released" is a more neutral word for the same thing.