The logo sketches were most likely Stan Lee's work. Some of the other earlier FF pages indeed have rough (i.e. really bad stick figures) "layouts." It's possible that someone else could have done them, but A) it almost certainly wasn't Jack and B) not many other people would've had access to the original art.
Now, the other possibility for the logos could be Sol Brodsky. He appears to have done some elaborate re-work on the Human Torch figures throughout that issue (at right, circled in red) to bring him more in line with the original Carl Burgos design. So Brodsky would have had access to the art boards, and he did create the "Fantastic Four" logo that shows up on the cover.
As for the sketches showing up on page 16 instead of 7, that doesn't strike me as particularly odd. Jack wouldn't have only turned in one or two pages at a time, he turned the art for an entire story at once. So when Stan (or Sol) opted to doodle some new logo ideas, he could've grabbed any page from the story at random.
In Pure Images #2, Greg Theakston goes back to the original art and inked some of the pages onto new boards using some of Jack's original pencil lines. Jack's original FF logo was an interlocking "FF" similar to one of the middle ones on the far right of that page. (Circled in red at the left. The final is circled in blue.) Further, the costumes included masks for everyone, hence that really weird close-up on Sue's face when she first steps out in the new costume. (See below.)
Fantastic Four #3
Over at the Kirby Museum, Rob posted some original art from Fantastic Four #3. What's interesting is that on the back of page 16 is a series of sketches for possible logos to use on the FF's costumes. I sent off an email to Rob with some additional ideas/comments, but I thought I'd repeat it here because... this kind of stuff interests me. Here, though, I've also included some highlighted art for clarity.