Never a better born Englishman

She is a bit "trying"="annoying"? obnoxious?

I think it's closer to annoying and irritating than obnoxious, but you have the idea.

Again, questions about single words are easily answered using a dictionary. This site is for phrases.

"These are the times that try men's souls." (Tom Paine, American patriot.) And these are the women who will test your patience. Same verb, so to speak. "Trying" is a so-called verbal, in this case a present participle used as an adjective, the original verb being "try = test". SS

SS - I'm not sure I'd call Paine an American patriot. Here's a description from one website;
[Thomas Paine] was detached from local patriotisms and national interests, a delegate at large in the cause of human rights, concerned with spreading the gospel of freedom in all lands.
And here's a strangely logical sentence from another; But his radical views on religion would destroy his success, and by the end of his life, only a handful of people attended his funeral.

The fate of his body is a mystery!
I will dance to Tom Paine's bones, dance to Tom Paine's bones,
Dance in the oldest boots I own to the rhythm of Tom Paine's bones.

If we are talking about the same Tom Paine, he was English. 'The Rights of Man' and so on.

DFG