1) He is more street smart than Lex.
2) He knows Superman better than Lex and also got to know him as Clark Kent.
Superman doesn't consider Batman to be the most dangerous man in the world for no reason.
Wrong on Luthor. Lex CAME from the streets, he, unlike Wayne, is a self made billionaire. Luthor is
very street smart, and probably more so than Wayne; for Lex the streets were his natural habitat, for Bruce they were something he studied.
Superman would not underestimate Batman and I am certain he considers him in the top 3, if not
the most dangerous man on the planet.
You raised a good point about the Luthor/Batman comparison, and one I've stressed before in older arguments: Any argument for Batman winning is one for Luthor winning. Aything Batman could do to beat Superman, Luthor could, and
would do, times 2. Unlike Batman, Luthor hates and genuinely
wants to kill Superman, realistically, Batman and Superman would never be fighting - the only way I ever see that happening is of one clashed with the other's ideals so much that they would feel like that were the only way to keep the other out of the way, it would never be with the same hate/passion that fuels the Luthor/Superman or even Joker/Batman dynamic, it would simply be if the one got in the way of the other.
The only way Batman beats Superman is if Superman really just
doesn't want to win that badly. Superman's not using his powers to the full extent of his abilities, not giving 100, even 70%. Even with the best laid plans, Batman is a goner. How many times have we seen this in previous fights with Superman and other characters? The only way there even is any sort of "fight" is because Superman hasn't quite put his foot down completely yet, so they go a few rounds - all because he doesn't really want to hurt anyone. Once that barrier is crossed, Superman is unstoppable, and will tear through even the best laid plans like paper. Anyone ever check out the DC: New Frontier prequel one shot where Superman and Batman meet for the first time? Batman spends MONTHS and TONS of money setting traps for Superman, his best plan at the time slowed Superman down for like 13 seconds for Batman to get a "head start", right before Superman caught up with him.