Don't forget periods at the end of sentences! 1) They will not give up the smallest particle of their liberty. FINE - a little rhetorical/poetic. 2) Mr. King is the best teacher in the whole world; his classroom should be the paradigm of all classrooms. GOOD 3) The presence of thousands of mendicants in every urban area is a sad commentary on our national priorities. GOOD, except many US speakers may not know the word MENDICANT, even though it's a good word. In the US people are more used to terms like: homeless people, unhoused people, beggars, panhandlers. Which term to use has become a subject of some controversy. 4)He took her arm, linking her hand through the crook of his elbow. GOOD 5) Barney's jokes were so insipid that no one in the room managed to force out so much as a chuckle. GOOD 6) Bribery is indispensable in our country to move your files in any government office. Bribery is the phenomenon of bribing. Good job, only minor alterations.

4 and 5 are grammatically correct. 4 is natural. 5 could be ok in context.