How did John's dead wife appear and phone John in "The Mothman Prophecies"?

TLDR:

To all your questions, we really don't know the answers. The movie is based on a book which has none of the information (he isn't even married in the book) and the film's script never explores this.

Ultimately, it's not relevant whether it was really his wife, whether it was Indrid Cole i.e. the Mothman, or whether it was something else. As the end of the film shows, what was much more important was that he left the Mothman and all the hunting for it behind, and rejoined the living with Connie (as he would never be able to solve the mystery).

Answer:

Firstly, the film is based on a novel of the same name which you may be interested in checking out.

Unfortunately, whilst a novel (as a primary source) usually helps, in this case it does not. The film has been "Hollywood-ised" and many of the events from the book are changed. For example, the character of the sheriff in the books is actually a journalist. John never worked for the Washington Post (so was most certainly not a star reporter). And most importantly of all - in the books he is never married.

So, the source material is no good to answering your question - we have to rely solely on the film (and therefore draw some subjective opinions from them).

With that aside, the only answer to your queries can come from the films.

Firstly, it's vital to understand that the film is collection of paranormal events, that all blend somewhat in to each other. However, it's incredibly difficult to derive any meaning from them. The film doesn't have a single story that begins and ends. Instead it has a snapshot of John's life, with no resolve to the Mothman story and no resolve to the mysteries.

For example, we have no idea what Indrid Cole i.e. the Mothman was/is. We have no idea why some people saw him/it and others didn't. We have no idea why their eyes were burned, or their ears bled. We have no idea why/how Gordon contacted him when he should have been dead.

The best "explanation" we get of the Mothman is from Alexander Leek, the novellist, during this conversation:

Klein: So you're telling me these things do not exist?

Leek: Oh, they exist. All kinds of things exist around us that we never see. Electricity, microwaves, infrared waves. You know, these things have been around forever. They show up in cave paintings. They're a normal condition of the planet. They're just not part of our consensus of what constitutes... physical reality.

Klein: What are they part of, then?

Leek: You're asking for an explanation... for something... that can't be explained rationally.

It's not exactly a scientific explanation.

So, with regards to your questions:

She (John's dead wife) contacted John by phone and convinced him to go to hometown to meet her. How does this happen and what is the explanation behind this?

To be exact, John was told by message to go home and see his wife. He never was phoned by her. He heard her voice on a recording, but not by a phone call. How this happens, or who sent the message is unclear. The explanation behind it is totally unclear too.

Also, I would like to know the significance of these incidents and the reason behind making John go to home town. Was it actually John's wife's ghost and trying to save him from the tragedy? Or, is it by the mothman or the Ingrid Cold?

Again, we have no idea. Given the fact his wife is dead, it seems likely it is Indrid Cole, i.e. the Mothman. in the script, this exchange occurs:

Klein: His name was Indrid Cold.

Leek: It's perception,John. They appear differently to everyone. A voice, a light, a man, a monster. If your friend thinks it's God he spoke to, he's off by more than a few degrees.

So it seems most likely that the voice is from the Indrid Cole, i.e. the Mothman because that's apparently one of his abilities. Similarly when the sheriff sees his dead wife, it appears she is seeing Indrid Cole, i.e. the Mothman because he can make himself appear as whatever he wants to.

The reasoning behind this is not clear though. Perhaps he wanted to make John happy. Perhaps it was a test of some sort. Perhaps he wanted him away from the bridge, so he couldn't intervene.

Conclusion:

We simply don't know. It's a very fun, very unusual supernatural film which has little grounding in reality. Most of the information from the book has been changed (and even if it weren't, the book is also highly subjective too). Given this, we don't know.

The only thing we do know is that John threatened to let the Mothman take over his life. But when the sheriff called him at his house, he made the decision to not pick up the phone to his wife, or whoever it was ringing, and return to normality. He drove to Ohio, saved her, and at the end of the film it appears like their supernatural occurrences are behind them.

Think of the end of the film says:

Klein: I left DC right after you called. You did call me, right?

Mills: Yeah. Yeah, I called.

Earlier in the film, there were a few instances when phone calls were supposedly made, but the caller denied it (and it was Indrid Cole instead). But not here. This is real, and with that the mystery of solving who Indrid Cole i.e. the Mothman is becomes irrelevant because, as Leek says, they'll never know.

Apologies if this doesn't "answer" your questions! I simply don't think anything beyond this can be provided.