Desert dream sequence.

Who did Batman 'take away' from Superman?

Lois Lane. Most likely following a similar path as the Injustice storyline, but with Darkseid instead of the Joker. In Injustice, Joker tricks Superman into killing Lois Lane and setting off a nuke in Metropolis.

Who are those flying objects?

Parademons. Darkseid converts creatures into parademons to aid help capture the worlds he is after. Superman is his greatest trophy since he is also the one of the most powerful.

Why is Batman even having this dream? Why does the Flash appear out of the computer when Batman hasn't even met the Flash?

We won't know anytime soon, but we known that Flash was physically there. That was not a dream, but rather a vision of the future where Superman is turned by Darkseid and the Justice League has failed. We are likely going to see something similar to the Flashpoint storyline where Flash goes back in time to prevent that change from happening but he went to the wrong time instead. That is also likely the cause of Batman having those visions.

This message was not communicated to Louise Lane, yet as soon as Batman mentions it to Wonder Woman, Louise Lane reacts and retrieves the kryptonian spear.

She was there when Batman fought Superman. She knew that the spear can affect someone of Kryptonian origin. She likely figured it out that Batman brought the monster exactly where he fought with Superman for some reason. The area being empty was only part of the reason.