These "plot hole" issues...all this talk of nitpicking. I'm sure somewhere in the film there's a plot hole. But "they could have made another choice" is not a plot hole, and many of the issues people are bringing up are certainly not plot holes.
Here's the thing about these kinds of issues.
When you start saying things like "Wait, a character COULD HAVE done this or THIS would have been better" then pretty much any movie ever made falls apart. It's not unique to BVS. Something else working better does not equal the thing that actually exists not working at all.
Characters in movies and stories don't always make decisions based on the most obvious and prudent thing to do...they make decisions that lead to dramatic potential. That's the nature of storytelling itself. Making the bold choice to lead to something interesting and hopefully unique. By the same token, authors often hold back certain details to increase suspense, surprise, etc.
With that in mind...
Why did Superman go to fight Batman instead of saving his mom?
Because Luthor is quite clear that if he doesn't, he will have Martha killed.
There's nothing established in this world to suggest that Superman could find his mother beyond the shadow of a doubt.
Narratively, because having Batman fight Superman, gain a grudging respect and understanding, and then have Batman save his mother is grand fashion while Superman confronts Luthor dramatically is more narratively and visually interesting.
How did Superman know Lois was in trouble in Africa?
Who cares? Doesn't really matter. Since when has how Superman shows up in the nick of time to save people ever mattered? You need a reason?
1. They live together and he knew her undercover work
2. He asked someone who knew.
3. A wizard did it.
Pick one.
Why was Batman just waiting on the roof for Superman and no one noticed he was there?
Who says no one noticed he was there? Why is this remotely relevant to the actual narrative?
How did Lois know the spear would kill Doomsday?
1. Because it's made of a substance that she saw weakened her Kryptonian boyfriend.
2. Because she can tell, based on the "activity at the Kryptonian ship", followed by the thing that can wipe the floor with her superstrong boyfriend and shoot HEAT VISION that it's likely Kryptonian.
3. Even if she's just desperate and GUESSING that Doomsday is Kryptonian and that it will harm Doomsday, it's still worth a shot, isn't it?.
Why did Batman make a spear instead of bullets when he uses bullets against thugs?
Symbolism. It's most likely meant to represent the spear of Destiny.
How did batman aknowledge Superman was no longer a threat?
He didn't. Superman proved he was no longer a threat by dying to stop Doomsday, and then dying. Dead beings usually aren't a threat.
How did Wonder Woman just show up out of nowhere?
She's a superhuman. She seems to be wearing motorcycle leathers in the airplane as she's leaving, so my guess is she took a motorcycle. (Shrug)
Why is Batman asking Superman if WW is with him?
Because Superman wants to know. Neither of them know who she is, where she's from, or her allegiances.
How did Lois get to the top of that roof where Supes and Bats were fighting?
She took a copter to that building, then took the stairs, or possibly an elevator. Or maybe a ladder. All three of which buildings tend to have one of if they have roof access.
Why was Jack not evacuating Wayne Tower until Bruce told him to?
Assuming that's the case, maybe he was in shock. But there's nothing to suggest no one wasn't already evacuating, just that Bruce saw the urgency of the situation. They could have been evacuating, and people could have been cowed and awed by, you know, the giant spaceship destroying nearby buildings.
Why did Lex create Doomsday if he thought Batman would kill Supes anyway?
It's pretty clear Doomsday was a backup plan. Since he's utilized as just that.
Why is Doomsday more powerful than Zod when mixing Lex's blood would inherently make him weaker?
I didn't realize we could say with any certainty how Kryptonian genetic manipulation worked. And let's say human DNA DOES weaken him; does it matter if he's weaker if he's going to evolve to be stronger each time anyway?
What the hell is Flash talking about?
There are several theories on this. The most obvious is that Flash is from the future, Superman has been a threat there, and Lois is believed to be the key to stopping him before it's too late. He also delivers a fairly straightforward "find us", referring to people like him. But even if you don't know or catch all that, that's fine. Bruce doesn't know what to make of it either until a later scene puts some of the pieces into context.
What was the point of the Knighmare sequence in the context of the story?
To show a visual representation of Batman's fears, IE a possible future.
Why didn't Bruce research Supes at all?
One of the first scenes with Bruce in the cave shows Alfred accessing stored satellite footage of Superman and data regarding him. Based on that, I'd say saying Bruce didn't research him at all is a stretch. We just didn't see that researching Superman. But we cannot assume he did not.
What would researching him have changed? He knew Superman was saving lives. He was angry and afraid about Superman's potential to do otherwise. Batman's actions in this film are not entirely rational.
