One thing I did notice is that there is an 1.5 inch gap in the door seal at the bottom near the front. Is it possible that the rain runs down the seal to the bottom and is then drawn into the floor by capillary action? Is that gap normal or should the two ends if the door seal meet?
It's possible. It's also possible for water to come past the door seal up higher if the seals are original or older and have flattened out a lot.
It's also possible to have cowling leak into the cab that isn't rusty. It's far more common if rust is present, but it's possible for the sealant at the cowl seem to dry crack and leak. Water leaks are pain to track down in my opinion, but it's got to be done.
A friend of mine is a glass guy and taught me this trick. It works way better than spraying a hose at the windshield and hoping to see a leak. This is going to take two people, or at least two people make it a lot easier. If you have sprayway glass cleaner on hand you can use that, or mix up a spray bottle off soapy water that will bubble when you shake it. (Windex is not foamy enough, nor is most washer fluid) spray all around the outside of the glass seal. Use your air nozzle, and hold it about 6 inches from the seal and slowly spray air all around the seal. When you get to the spot it leaks at your friend on the outside will see the bubbles. You can do this on the door seals as well, but it is possible to get a false positive on door seals because sometimes those can leak air but not water. Just be careful not to get the air to close to the seal as if you applied it directly to the seal your could blow it out.
Also take some silicone spray. holding it just a few inches from your door seal apply it all around the seal liberally. Have a rag handy to wipe up over spray but let it stand heavy on the rubber seals. After a few minutes the seal will have absorbed the silicone and it'll look like it is starting to come back into the round shape it should be. Give it several applications, letting it soak in for a few minutes between coats. Not only will it help the door seal up better, the seals will look much nicer and newer afterwards. Don't try to use silicone grease it's got to be thin spray on stuff so it'll soak in. As far as I can tell this does no damage to the seals long term and I treat all my car door seals like this regularly.