You have asked more than one question so I have edited it down to only one question, per enotes rules. However, this is a very interesting topic. Of course, irony comes into its own in this terrifying short story which has such a sting in the tail at the end of it for its readers. For me, the biggest form of irony that operates in this short story is dramatic irony, but dramatic irony with a difference. Normally dramatic irony operates when the audience and one (or more) of the characters knows something that another character or group of characters does not. What is fascinating is that in this short story, all the characters know what is going to happen - it is us, the audience or readers, who are deliberately kept in the dark until the very end. So therefore, the story achieves this surprise through a combination of dramatic irony and then situational irony, for we are surprised by the ending greatly.
Of course, looking back at the story after the first reading, we are able to pick up clues that we glided over before. For example, the reference to stones has a new, chilling meaning:
Bobby Martin had already stuffed his pockets full of stones, and the other boys soon followed his example selecting the smoothest and roundest stones; Bobby and Harry Jones and Dickie Delacrois - the villagers pronounced this name "Dellacroy" - eventually made a great pile of stones in one corner of the square and guarded it against the raids of the other boys.
Of course, on first reading, we think that this is just some kind of childish game, and never would we suspect the real reason why they gather stones. Equally, Mr. Summers's speech about the Lottery and its importance, and why it should be carried on in spite of other villages abandoning it also assumes new importance.
The real sting in the tail comes at the very end of the story:
Tessie Hutchinson was in the centre of a cleared space by now, and she held her hands out desperately as the villagers moved in on her. "It isn't fair," she said. A stone hit her on the side of the head.
As the story closes with the village stoning her to death, we are left in a state of shock produced by the irony employed in this excellent, masterful and chilling tale.