- What is the significance of a #0 issue?
- Are they considered a must-read as part of the series they're linked to?
- How common are they?
- Why does the Avengers vs X-Men #0 cost twice as much as issue
An issue #0 is usually a back-story issue that covers information the reader needs to know in order to continue onto the event.
An example that comes to mind is the Blackest Night issue #0 which basically gave a background on each of the colored Lantern Corps in the Emotional Spectrum and showed a few stories on why they are significant to the plot of the overall event. The issue sets up the story of who the main characters are for those that don't know and why they are starting the journey.
The reason why they aren't called issue #1 is simply because the story of the event in question hasn't started yet and is merely being set up for the readers. I HIGHLY recommend that if you are going to read an event and an issue #0 is present that you read that first and foremost.
And for your question on price, they are usually more expensive because they are much bigger. Blackest Night was I think 55 to 60 pages in length (not counting ads) while most comics by DC are 22-24 pages in length (again, without the ads). There's also expected collectors value of issue #0's which could drive up the initial cost. Also, some Issue #0's are limited edition, have different variant covers, have special tie-ins and addons included with the issue, etc, which all could drive up the cost of the issue.
The first #0 issue that I'm aware of was Zap #0, published in 1968. This was the material that was meant to be published in Zap #1 however the artwork was stolen prior to the release of the book and Robert Crumb produced new material that was included in Zap #1 instead. After two issues of Zap were released xeroxed scans of the original material meant for #1 were used to create an issue #0. As far as I'm aware that is the first zero issue and was probably the inspiration for the first of the #0 issues that became a trend in the 1990s during the speculator boom.
A zero issue can be used however the publisher wants to use it. For a series like Francesco Francavilla's Black Beetle, the #0 issue was reprinted material that was original published as a three part story in the anthology series Dark Horse Comics Presents. In that case it was material by the same creative team as issue #1 of the first Black Beetle mini series and was relevant to the story if not essential.
In the case of DC's Zero month from a year and a half ago the books were published as year after the first issue and even after the first collected books/trades had been released, so the zero issues would not be essential to what happened in the early issues. In some of the cases there were different creative teams working on the books and in many cases the back story that was being told directly related to the upcoming issues of the particular title, not to the earlier issues that had already been published.
In other cases an issue #0 is released as a promotional issue to generate interest in the story for when issue #1 is released. There is no hard rule to how these can be applied. A good indication to whether it applies to a story is to check to see if it was included in the trade or if it had the same creative team. Many of the #0 issues are junk and the series holds on their own better if they're ignored. Others are essential and others still add some context but aren't essential. The ones like the Black Beetle #0 that I mentioned are just a nice way to get a piece of a story without having to buy all the unrelated supplemental material that was included with it in the anthology. Do your research and try a few out. You kind of figure out the formulas of the different publishers as you get a sense of them.
As @OghmaOsiris has said, issue #0 provides back story for whole series. It introduces events and characters to readers which would matter in upcoming comics.
Particularly about issue #0 of Avengers vs X-Men:
It has set up the crucial roles of the Scarlet Witch and Hope Summers before their allegiances come into questions in upcoming comics. And, its very important before Avengers and X-Men go head to head.
As I have seen in ComiXology app, its cost $3.99 is same as of issue #1. But, its costly if you calculate cost per page. Its reason can only be provided by marketing department. I suppose, it has to do with last month's launch boom of this new mind - blowing series. They took advantage of it to clear our money. But, its just a guess..