To succintly answer your questions (1), (2), (3) (not necessarily in this order):
May wants out of the EU, but she wants out on the terms of the draft deal she negotiated with the EU... which is lacking support in her own Parliament (but the MPs couldn't decide to replace her as PM either; why that is is an interesting separate issue).
May and the UK Parliament did trigger (years ago) all the legal steps needed for the UK to crash out the EU with no deal... in the next couple of weeks (a little vague because extensions to the deadline are possible).
A no-deal exit is however not the Brexit that most UK MPs want, according a recent non-binding vote they took, which is basically the only thing they could agree on recently--that is what kind of Brexit they don't want. So basically their hope-for-the-best thinking (triggering article 50, which has a deadline to finish exit negotiations) is coming back to bite them in the sense that they'll (collectively) get their least favorite version of Brexit... by default.
For more background why no-deal doesn't look good (to most MPs), let me try to explain a few aspects of Brexit. Yes, I'll be oversimplifying by necessity and I'll make [possibly faulty] analogies with the US because the OP hails from there:
Getting out of the EU just by article 50 (which was triggered by PM May two years ago, following a [non-binding] UK referendum) without any deal is like scrapping most of your trade agreements (big understatement). Even if the EU were just a trade agreement, it's often preferable to change one rather than scrap it. Witness Trump's actions in this respect: he scrapped the non-yet-going TPP, but for most other, NAFTA, etc., he renegotiated them with some but not huge changes.
The trouble is that there's no consensus in the UK parliament what a replacement deal should look like, even though PM May did hammer a draft deal with the EU. Compare with NAFTA's replacement having trouble in Congress. Trump gave up threatening to just scrap NAFTA. Back to EU, the looming problem is that Article 50's legal two-year deadline to reach a deal means there's an automatic scrap about to happen. A short extension to the Article 50 default-no-deal-deadline has been agreed mutually agreed with the EU, but the EU also got tired of the indecision in the UK parliament as well.
But the EU is not just a trade deal; in a certain way it, is was also a part of a peace agreement in Northern Ireland. The de-escalation in Northern Ireland (Good Friday Agreement) was largely possible because both the UK and Ireland were part of the EU's single market, so the two countries could make invisible the border that the Northern Irish republicans fought against (the latter because they advocated a union of Northern Ireland with Ireland). So dropping out of the EU is also like scrapping a part of peace deal. Imagine a US state [legally] seceding from the rest of the Union, while a part of secessionist state's population threatens with civil unrest or even domestic terrorism if a border is actually put in place after the secession...
So the draft deal May agreed to "gave in" to the troublesome guys [Northern Ireland republicans] by allowing Northern Ireland to stay in a customs' union (so no visible border) with the EU until someone can figure out a better solution (so it's a open-ended concession/delay); this is the [in]famous "backstop". But this backstop pissed off the "take back control of our borders" guys from her own party coalition (including a Northern Ireland unionist party--unionist here meaning they want [to stay in] union with the UK--that is actually critical to keeping May's government in power and also to the ratification of a deal).