We cannot extend unilaterally. It needs EU's consent and they've already stated that: 1) it won't be free - £1bn/mo has been mooted from various sources, 2) there would be clear legal strings attached such as a requirement for a 2nd referendum, 3) extension would only be allowed up to the day before the EU Parliament elections on 23rd May.

Ah I thought it was extension but from other comments it seems they can just cancel Article 50, my bad. That requirement for a second referendum might be a good thing though, seems like that's the only way out politically (to be forced by an outside force) given how many of the heads party have said a second referendum would tarnish the democratic process (which is BS to me but that's their stance).

Why would the EU allow the UK to extend this one-foot-in-one-foot out nonsense? It's not clear that any deal it is willing to offer will get parliamentary assent.

Because European courts have decided that Article 50 can be unilaterally retracted.

I was under the impression that it can be only unilaterally "retracted" (i.e., cancel brexit), but extending the negotiation required EU's agreement, which already stated that they wouldn't unless there's a concrete plan?

Yes. In practice the UK would have to ask the EU for an extension, or article 50 could be retracted and resubmitted, achieving the same thing. There would be political barriers to doing that so the EU does have some bargaining power.

The EU doesn't want the UK to leave. It would damage their project massively. It might even unravel it entirely if some other countries decide to follow in the UK's footsteps.

If the EU doesn't want the UK to leave, then its best option is the give the UK as much rope as it needs to hang itself, and pray that the UK comes to its senses, instead of giving it more time to negotiate a Brexit that will actually get parliamentary support.

"We'll hold another referendum, and cross our fingers that this time, it will vote remain" isn't exactly satisfying to the EU.