FC said just the opposite, Earth fully recovered in a half century. Suggesting that any chaos and "mad max" was probably few and far between.
I doubt Picard would send his crew to a region with such a society.
From the script:
DATA: According to our astrometric readings we're in the mid twenty-first century. From the radioactive isotopes in the atmosphere I would estimate we have arrived approximately ten years after the Third World War.
RIKER: Makes sense. Most of the major cities have been destroyed. There are few governments left. Six hundred million dead. No resistance.
Plus Cochrane and his friends lived in shanties and tents, who's to say there weren't any dangerous elements around that showed up regularly.
Sure, after the Vulcans showed up Earth recovered within 50 years (some earlier episodes of TNG suggest that it was gradual and not all places on Earth recovered at the same pace)
And 50 years is a long time.
I agree that Picard would have chosen a place close to "safe" regions, but there's still a risk. If it's between tents next to a missile silo in Montana where some remnants of the Mad Max people might bother me (no matter how small that chance) or tents on a tropical island where I can be pretty sure that they wont, I'd take the island (if it was tropical/subtropical, you are right in saying that we don't know, But then again, why would Picard have sent them to a barren piece of rock in the south Pacific?)
That the E-E would be dedicated to combat is a bit dubious when we note that she was specifically excluded from combat against the Borg, and never seen fighting the Dominion, either. Yes, Picard may have been considered untrustworthy, or more valuable in other roles, but that wouldn't have stopped Starfleet from sending the ship (and placing Picard aboard another), or from employing sister ships. Perhaps it's the Sovereign class that is the failed experiment in the end?
I mean in the movie they discuss that the Enterprise E is the ship best equipped to combat the Borg threat, but are relegated to the Neutral Zone because Starfleet Command doesn't trust Picard in this situation. And that's part of the reason why they then go fight the Borg anyway.
As for the civilians boarding later, you have a point there. Theoretically they could also have taken the time to unload the civilians and non-vital personnel at a Starbase or Colony before they went to fight the Borg, like the Odyssey did on Deep Space Nine before engaging the Diminion Ships. So Mott and Guinan and all the little babyleins were sitting around on some Space Station somewhere between the Neutral Zone and Earth while the Enterprise was battling Borg in the 21st century.
Truth be told that might have happened during, at least, Nemesis as well. No need to take the kids to battle the Romulans.
Now I want a spinnoff movie about Guinan and Mott babysitting the Enterprise's children during First Contact/Nemesis.
Highly capable of combat yes, but it's hard to imagine that big ship just sitting around for months on end waiting for a combat assigment to respond to.
I didn't say that the Enterprise E was there just for combat related missions, just that the ship was designed with the increasing threats towards the Federation in mind, and would (that issue with Picard and the Borg notwithstanding) be among the first to be called on for missions like that, while there's likely ships who are more geared towards other missions as well.