From the sound of it, there are likely two problems affecting your Transit. The first is a binding gearshift cable and the second appears to be with the gear activator solenoids.
Here’s what happens when you normally shift your van. Let’s say you select drive and slide the gearshift selector to the D position. There are a couple of issues that should happen here. As you pull the vehicle out of Park, the dog collar securing the output shaft releases and allows the transmission to work correctly. Once the dog collar is freed up and parked for normal operation, the torque converter spins up and starts transmission fluid flowing threw the transmission. At this point, depending on the input from several sensors, your transmission chooses the gear it wants to start off in. In a five-speed, it means your Transmit may start off in first to fifth.
The actual choice of gearing is governed by the solenoid. There are usually two solenoids located on the valve body, one internal and the other external (some multi-geared transmissions may add another). Depending on which relays are set to fire by the solenoids, you then have the final gearing choice that is sent along to the clutch/band combinations.
In theory, it is quite easy to do. The reality is far different as the solenoids must fire at the right time; the valve body has to be clear and free of debris so the proper check valves allow the right gearing choices and fluid route. Now, if one of the solenoids fails to fire for one reason or another, not only will your Transit shift in the manner you describe, but it will also likely go into limp-home mode, which you also touch on in your question.
If your transmission goes into, say, first and reverse and no other gears, your transmission is in limp-home mode because the engine control computer (ECC or ECM) has received a code from a sensor or other electronic device telling the ECC there’s a problem.
So, now we have all of the possibilities on the table. The good part, for at least two of them — the binding cable and solenoids — is that they are relatively quick and inexpensive to fix. If you have to repair the transmission select cable (a gearshift cable within a runner) because it has stretched beyond all belief due to years of wear), as well as the solenoids, you are looking at a parts cost of about $450 and labor of another $300. This is far better than installing a rebuilt transmission for $3,400, isn’t it.
If, on the other hand, the problem is more serious — an error code and limp-home mode — seem to indicate that it might be, then you are looking at the diagnostic and locating the problem. I suspect that your ultimate problem (number four) is, like as not, in the planetary gearing itself. The only way you are going to find this out is a major diagnostic teardown which will cost on the order of $1,450, plus any replacement parts. In this instance — only this one — I’d suggest a rebuilt transmission because your ultimate costs will be less. Please let me know how this works out.