It always struck me as completely nonsensical that Kiki regains her ability to fly but not to talk to her own familiar. I get that they were trying to insert a Growing Up Sucks thing into it, but it just strikes me as inconsistent/violating the rules of their own setting.
Furthermore, this idea that witches-in-training need to be able to talk to their familiar, but full witches don't is a bit odd. Furthermore, if it's common that witches cease to understand their familiar when they get powerful enough, why is Kiki shocked that this has happened? Surely her mother explained the major steps in this process before Kiki left!
I never gathered that Kiki completely lost her ability to talk to Jiji. When he jumps on her shoulder at the very end of the movie, it's obvious Phil Hartman does the 'me-YOW!', where when she couldn't understand Jiji before he sounded like a real cat and made cat mewls. I saw it as Jiji able to understand and talk to her, but he was just making a joke that time, and we just didn't get to hear from him again after that.
In the dub, Phil Hartman also had Jiji say "Kiki, can you hear me?" before he jumps on her shoulder and meows. In the Japanese version, it was just another regular "meow" but they decided to change that for the dub probably because it made more sense and because the whole losing her ability to communicate with her cat wasn't in the original book to begin with.
In-universe logic is secondary to storytelling in Hayao Miyazaki movies. In this case, Miyazaki is delivering An Aesop about how in order to grow up, you have to leave behind the familiar things of your childhood. So the radio stops speaking Japanese, Kiki loses her mom's broom, and she can't understand Jiji anymore. Jiji represents Kiki's childish self, so her inability to understand him after regaining her flying powers is symbolic of her having grown up. I think the 1998 Disney dub restored Kiki's ability to understand Jiji in order to give American kids a happier ending.
How many different English dubs ARE there? In my original VHS of the movie, it was always Phil Hartman making the meows, even when they couldn't communicate. Now I have the special edition DVD and not only does Jiji sound like a regular cat AFTER she gets her powers back, but they also cut out a lot of his  lines. SO CONFUSING.
There are three. Streamline did a dub not long after the movie came out. This script was used for the subtitles on the Disney releases. Disney re-dubbed it in 1998 for a DVD release, with some new music and lots of extra dialog from Jiji. This dub was then edited to be closer to the Japanese original for the 2010 rerelease.
At the end of the story, can Kiki still understand Jiji? Original book: Yes (she never loses this in the first place.) Miyazaki's script: No. Streamline dub: No. Disney 1998: Yes. Disney 2010: No. Can this get more confusing? NO!
So Disney, in making a change to the dub, actually brought the movie closer to the source material?
Miyazaki's statements aside, the mere fact that Jiji doesn't talk to Kiki in the final scene isn't proof that he can't. In fact, Jiji never talks to Kiki when others are present - only when they are alone.
Why do so many people randomly wink at Jiji for no reason? Asano, Asano's husband, Ursula. Is this some weird in-joke?
Maybe they're just showing awareness that he's sentient, as opposed to a regular cat?
Is there an actual reason Kiki is so mean to Tombo at first? I don't really see him doing much that warrants her being so rude.
At that point, Kiki had just had the run-in with the traffic cop. She's scared that she'll screw up her training and she'll have to go home or something. She's also embarrassed. But she's too disoriented to communicate any of that directly, so she just gets mad instead. Some people do that. Kiki makes a similar mistake when Tombo invites her to come along with his friends in the car (just after the bicycle scene). She feels awkward around these other people, but rather than just saying so she simply walks away without explanation. She's not very good at communicating negative feelings.
Also, Kiki's from a rural village and may genuinely be slightly shocked that a boy would speak to her without an introduction.