My take

Overall: These are visuals meant to exaggerate the chaos in a disaster movie's control room much like the various people panicking in it as well as literal s#!% hitting a literal fan. But more specifically:

The Spear: As commented above, I always felt this was also a callback to the stock-footage indians that were chasing Kramer in the end of this previous scene:

The Watermelon: I always felt this was a comical visualization of McCroskey words (note that it falls when he says 'right down to the ground' and matches his gesture) and a juxtaposition suggesting that the plane will crash in a splat, rather than land, on the ground.

It's possible to overthink the gag - they are both just cynical 'peanut gallery' jokes. The spear is suggesting that Kramer meant a REALLY long time ago. And the watermelon is a suggestion of how 'talking him down to the ground' is going to go - a big smash and lots of red everywhere.

I don't know if this will satisfy you, but it's about as official an answer as you're going to get.

TL;DR: They're just bizarre jokes.

On the DVD commentary, the writers and producer talk about scenes with Robert Stack and Lloyd Bridges, and they mention that the jokes don't always have anything to do with the action (these are my best guesses as to who's speaking):

David Zucker: He was a really nice guy, Bob Stack... He really understood it, too. One of the first read-throughs, Lloyd Bridges had a lot of questions about the script. He said, "Now why would my character say this? And why...?"

Jon Davison: He was actually trying to make sense of his dialogue.

David Zucker: Stack just laughs and he said, "Lloyd, there's a watermelon hitting the desk behind us and a spear flying into the wall. Just talk, nobody's paying attention to us."

Jerry Zucker: I also remember him telling Lloyd, he said, "Lloyd, we are the joke. Just keep going."

Jim Abrahams: Stack was so right on, he got it totally.

They confirm this later with another gag that has no setup:

Jim Abrahams: This is the famous mirror scene. I don't know if anybody really got it. It didn't get a laugh but it was bizarre. And so we decided to take that concept and expand it into Top Secret! and do a movie full of bizarre jokes.

And another pointless gag:

(Rex Kramer tosses a cigarette out the window and something explodes.)

David Zucker: Look—here's another joke we always thought was funny. The audience really didn't laugh, but we were hysterical the day we filmed that, we just thought "that's the funniest joke in the movie."

Jon Davison: And the watermelon, and the spear.

David Zucker: That's right, nobody laughed at that, either. But we left it in.