Happy Holidays to you and thanks.
From your description, it looks like your Toyo’s transmission has several problems, likely related to age and mileage. Though you did not give me a reading of the mileage, a good guess would be that it is in the range of 110,000 miles or so. Since you haven’t said whether you have ever changed the transmission fluid — like oil,it does have to be changed, though most manufacturers today claim that it is good for the life of your car. A good recommendation on changing the transmission fluid is every 50,000 miles or so. This assumes you drive 15,000-miles-a-year or a little more. To change the fluid, use the method we outline in: https://www.transmissionrepaircostguide.com/change-transmission-fluid.
If you do swap the transmission fluid, could you tell me its condition? Does if feel gritty? Can you see any metal filings in it? Does it have a burnt odor? Is it black or tarry? Also, does the transmission pan plug, which is usually magnetized, have metal shavings on it? If you find any of these conditions, it means your transmission has problems, not the least of which might be a bad turbine blade or metal-on-metal wear, perhaps in the clutchs or gearing itself.
From the sound of your problem, though, I think that the fluid change will just show you the way to these answers. Since your transmission is skipping and taking time to spin up, it sounds like you have a torque converter problem or, quite possible, a problem with the input shaft and its alignment, as well as the output shaft is alignment.
However, since reverse won’t engage, it is quite possible that the reverse solenoid is shot. You see, your transmission has internal solenoids that act as switches and that control the gear choice. The solenoids are located externally on the valve body and internally, as well. Any solenoid actuation problem will cause your vehicle to stop shifting.
There’s more, though. From your description, it sounds like you may also have a problem with the clutches and bands and, quite possibly, with the gears as well. And, there may also be sticky check valves in the valve body. A technician won’t know until he gets inside the transmission.
That is ultimately what will have to happen. I also suspect that you will be swapping in a replacement, rebuilt transmission because your current device is, I can’t say this strongly enough, shot. You are looking at a $3,000 to $3,200 bill, depending on your location.
My suggestion would be that if you like your Toyo, change the transmission. It is far less expensive than purchasing a new car. However, if you don’t car what happens to your car, then, save the money and use it on a new car. It will be years before your new car needs transmission work. And, if it does, when it is new, you have the warranty to pick up the expense. (At 16 years old, the longtime warranty on the transmission has lapsed.)
I’d really like to hear what happens, so let me know here.