Hello. I will really appreciate some help.

1)Do you think it's possible and natural to use "unsuccessfully" in the following position: "They have been trying to solve the problem in the new project, but unsuccessfully"?
2)Do you think the article is a must in the context: "He blamed the boss for the (?) intimidation of the whole staff"?
3)Is it possible to use "once" in the context: "He said that once the country joined the EU, it won a great victory" (meaning: joining the EU was an achievement, a victory for the country)?
4)Do you think it's natural to say "the President is supposed to visit France in May"?
Thank you very much for all your help.

1)Do you think it's possible and natural to use "unsuccessfully" in the following position: "They have been trying to solve the problem in the new project, but unsuccessfully"?

Yes, that works, but it's a bit awkward. This would be smoother: ... but they were unsuccessful.

2)Do you think the article is a must in the context: "He blamed the boss for the (?) intimidation of the whole staff"?
In this sentence, "the" is needed, yes.

3)Is it possible to use "once" in the context: "He said that once the country joined the EU, it won a great victory" (meaning: joining the EU was an achievement, a victory for the country)?
I'd use "when" instead of "once."

4)Do you think it's natural to say "the President is supposed to visit France in May"?
Yes, it's fine.