"Legal experts say the president has the authority to declare an emergency..."
I believe this point is disputed. Certainly, it would end up being dragged through the legal system for some time. And why not? At the heart of this dispute is a conflation of border security in general -- which, let's be clear, Democrats have no objections to funding -- and the Wall in particular, the case for which is far from convincing (to say the least).
It's also worth noting that the last 16 years have seen migrant flow across the Mexican border fall rapidly, by just a shade over 75% since 2001 (from a high of 1.6 million), and while it did rise last year, the general trend is hardly suggestive of a crisis. If this is a National Emergency now, then why was it not in 2001, when the problem was four times worse?
I'm surprised that Trump hasn't at least tried to declare a National Emergency already, though, and I think that rather speaks to the truth of it. It's a threat, trying to force the Democrats to buckle, rather than something he is wanting to do because it is objectively the right call.
Finally, on your last point -- this I can accept, as it's quite clear that the Wall, as part of his wider campaign, played a not insignificant role in his victory. So Trump does have a mandate for it. But the 2018 Midterm results gave the Democrats in the lower House an equally strong mandate to work against Trump. Both sides therefore have democratic legitimacy to back exactly opposite positions. No wonder both are sticking to their guns, and will presumably do so for the foreseeable future.
Trump has to accept that border security is possible to achieve without building his pet Wall. Considering that $5.7 billion is, anyway, only a fraction of the cost to build and maintain the thing, and considering that he definitely *did* suggest, explicitly, that Mexico would directly pay for it, then I think it's completely reasonable for Democrats to continue to resist him on this.