The car can jump without moving, this was shown quite clearly at the end of Part II. Sure, the circuits were scrambled and all, but it was otherwise just hovering in place. I always assumed the 88mph was just arbitrary, possibly a way of preventing an accidental jump since you'd have to intentionally get up to speed (and let's be honest, the DeLorean wasn't the most powerful car, especially the original version), so 88mph, while certainly doable, isn't something that would just happen by accident). Perhaps it was something built into the computer and 1955 Doc wouldn't know how to change it, and 1885 Doc wouldn't have been able to adjust it without another bigger computer to hook into it (although the real reason, of course, is because they needed a big action piece at the end of both movies).
I'm going to self-correct here, having just finished watching the film for the first time in years. The 88mph is not arbitrary, but apparently a vital component of the time travel process. When the car jumped for the first time with Einstein in it, Doc was happier about the fact that he had correctly predicted it would happen at exactly 88mph, so it wasn't anything he had specifically envisioned, it was just part of the process of making the car jump. Which then begs the question of how it jumped at the end of Part 2, unless it was simply that the entire flux circuit was overloaded.
The bigger question I had watching it this time was how he was able to remote control the car, including moving it in reverse, when it had a manual transmission.
Going back to the other question about the timing of the lighting strike, after Doc reads the flyer about the clock tower, he says that it occurs "precisely" at 10:04, and he seemed pretty confident about it. It's implied that they were indeed able to accurately determine the exact time of the strike, probably (as I mentioned earlier) based on the interior components of the clock. Unfortunately, this is purely a trick of the script, as the flyer prop didn't have any particular text on it, only the headlines were important and the rest was filled with the random drivel that all movie newspapers have.
As to how the clock was stopped by the lightning strike, being entirely mechanical in nature, Doc himself wrapped the electrical cable around the minute hand to secure it at the top before he slid down to the bottom to reconnect the other end. This means the electricity was also shunted directly into the mechanics of the clock, probably fusing them, in addition to what went down the wire to the car. I suppose you could argue about whether he always did that in every conceivable timeline, but you could assume that he did.
As to the OP, the mall's name isn't remotely hidden.. Zemeckis has the camera linger on the sign in both timelines, making it impossible to miss except for maybe your first time watching, since your attention is elsewhere.