Hello. I need some grammar and lexical help.

1)Is it possible to use the infinitive with "purpose/goal/objective/aim": "the initiative has a purpose/goal to improve the situation"
2)is it possible to say "the visit intends to demonstrate..." (or is "intend" used about people?)
3)is it possible to say "questionable/questioned territory" about disputed territories
4)which is correct,will or would, in the sentence: they belive the countries will/would hardly sign an agreement.
5)are "remain" and "stay" synonyms in the phrase "they remain / stay unshakeable in their position"
6)is it natural to say: "the remarks started criticism"

And one more question please: are such words as "however, nevertheless, though" separated by commas or does it depend on their position in the sentence?
Thank you very, very much for all your help.

#1 would be better phrased like this:

The purpose of the initiative is to improve ...

#2 - intend is used for people, yes. You could write this: The purpose of the visit is to demonstrate ...

#3 - It would be better to refer to questionable boundaries or boundaries in dispute.

#4 - Read these:
http://grammar.ccc.commnet.edu/grammar/verbs.htm#sequence
and
http://grammar.ccc.commnet.edu/grammar/sequence.htm
Since you have a simple present (believe) for your main verb, what should you have following it?

#5 - Yes, they are synonyms for each other.

#6 - I don't know how remarks can start anything. To me, "start" is more of a human activity. You could say, "The remarks resulted in criticism..."

However, nevertheless, though -- it depends on how they are used. Most of the time (but not always), they should be used in a compound sentence with a semicolon before them and a comma after. They are called conjunctive adverbs (and "though" can also be a subordinating conjunction):
http://grammar.ccc.commnet.edu/grammar/transitions.htm#transitions
Scroll down to the chart. If you give me your whole sentence I can help you better with these.

Hello! I'll be happy to help you with your grammar and lexical questions. Here are the explanations for each of your questions:

1) Yes, it is possible to use the infinitive form with words like "purpose," "goal," "objective," and "aim." So, you can say: "The initiative has a purpose/goal/objective/aim to improve the situation."

2) "Intend" can be used with both people and things. So, it is correct to say: "The visit intends to demonstrate..." Here, "intend" implies that the purpose of the visit is to demonstrate something.

3) Yes, both "questionable territory" and "questioned territory" can be used to refer to disputed territories. "Questionable territory" indicates that the territory's status is in doubt or uncertain, while "questioned territory" suggests that the territory's ownership or legitimacy is under dispute.

4) The correct word to use in your sentence would be "would": "They believe the countries would hardly sign an agreement." "Would" is used here to express a hypothetical or conditional situation.

5) In the phrase "they remain/stay unshakeable in their position," both "remain" and "stay" can be used interchangeably, as they are synonyms in this context. Both words convey the idea that they continue to be unshakeable in their position.

6) It is more natural to say: "The remarks sparked/came under criticism." Instead of saying "the remarks started criticism," it is more common to use verbs like "spark" or "come under" to indicate that the remarks caused criticism to arise.

Regarding your last question about using "however," "nevertheless," and "though" – these words are used as transitional adverbs to show contrast or introduce a contrastive idea. Their usage with commas depends on their position in the sentence. When these words are used at the beginning of a sentence to introduce a contrast, they are usually followed by a comma. For example: "However, I disagree with your opinion." On the other hand, when these words are used within a sentence to show contrast, they are not usually followed by a comma. For example: "I understand your point; however, I think we should consider other options." In this case, the use of a comma after "however" is optional.

I hope these explanations help you! Let me know if you have any more questions.