It's more than that, it's being "close" to a border (you can argue if 100 miles is "close") and not on a whim but reason to believe you recently crossed it.
I've heard (but can't back up) that any city with an international airport counts as a 'border' for this purpose.
Sure, but if you're driving in NM or TX or AZ, what constitutes that reasonable basis for a stop? As long as the car is legal, there can't be any, yet they still do roving patrols. The also stop buses and trains mid-transit (not at stations), which again, hard to conceive of why that would be necessary - if they know an illegal is on a bus/train, sure stop it and get the person, but asking everybody onboard - should be illegal.