I mentioned to my n’bor that my cat always hesitates at threshold and then jumps over the threshold in a bit of a panic when entering my condo from the hallway. She said her cat does that too and that it’s common, and that it is why people sometimes say cats see ghosts (ie, they fear something we don’t see). So, do most cats do this? (Not see ghosts, the trepidation over the threshold thing.)
Not the CelynCat, anyway. Statistical sample of one. 
Only one of our three cats goes outside. When we get home, he’ll trot over to the door, but he will not go inside. We’ll stand there with the door open, letting out all the A/C and telling him to go in, but nope. So we have to go inside ourselves and wait for him to howl that he wants in, and even then he’ll cross the threshold with some trepidation. We just thought he was stupid, but if other cats do it too, maybe you’ve got something there.
I used to have a cat. She wouldn’t jump into the room or over the threshhold though. She would kinda peek her head around the corner and scope things out. Maybe it’s a natural instinct for predator/prey type of stuff.
Cats seeing ghosts though, I’ve always wondered the lore about this myself…
I’ve seen wary behavior as cats go from room to room, but I think it’s because they’re not sure if one of the other cats is waiting around the corner to pounce on them.
I’ve seen the exact behavior you describe with the outside doors on the house. My personal theory is that outside doors don’t stay open long, and they don’t want the door slammed on their tail.
Our cats sort of oozed in and out through doors, making sure that there weren’t any goblins lurking on the other side. And ready to leap straight up, or sideways or backwards if there were.
Well I prop the hall door open. I used to think that it was because he didn’t like the looks of the black gym back or large pillow-cushion that I would use, but it doesn’t matter. He doesn’t do this for the threshold to the bathroom, which is more conspicious.
Huh. And here I was thinking it’s because cats want to see just how long they can get those dumb monkey’s cousins to hold the door open while they take their sweet time deciding “in” or “out”. 
I know that one of the games my cats play is hiding around the edges of doorways waiting to ambush another coming in.
It’s funny to watch, sometimes the hiding cat is not hidden well enough, and when it pounces out, has to sort of try freeze in mid air and look as nonchalant as possible.
The game appears to be ‘if you spot me I’ll pretend I was doing nuthin’ and ‘if you don’t spot me I’ll chase your sorry ass’
Sometimes the pounce creates a counter reaction where both cats jump upwards and land on the same starting spot.
I read once, years ago, this little trick. If you have a checkered tiled floor (say alternating square of black and white), take a two-by-four and paint it black and white like the floor - alternating sections of black and white, the same width as the floor tiles. Lay the board on the floor atop one row of tiles, with its colors contrary to the floor’s colors - black on white and white on black. Try to get a cat to step across the board. Cat won’t do it.
Unfortunately, I’ve never had the opportunity to test this for myself. I always assumed that the contrary pattern played some trick on the cat’s vision or something.
I think a lot of animals have a natural concern about changes in the texture and color of footing. This is obviously a useful instinct in the wild.
Also, she may have once gotten a static shock from a metal plate threshold (such as you see where carpet ends) and is now overly concerned about every kind of threhold.
I’ve been ordered by cats that owned me to open doors for them. They obviously didn’t fear thresholds.
I remember one amusing piece of home video showing a cat waiting patiently at a glass door. The glass had been removed and even though the cat watched people stepping through, she wouldn’t move until the empty door frame had been opened.
True. Horses and cattle won’t cross acattle guard that consists only of while lines painted on pavement.
Pigs on the other hand, being of superior intelligence, will stroll across a real, live cattle guard as if it were made for their convenience.