Am I correct that "The car that should be/will be/must be/can be washed is here" can be reduced to "The car to to be washed is here" no full stop here or to "The car to wash is here" if we (the speaker) are doing the washing?
That's a very long single sentence! Let's start with the four sentences contained in your first set of quotation marks:
1. The car that should be washed is here.
2. The car that will be washed is here.
3. The car that must be washed is here.
4. The car that can be washed is here.
They're all grammatically possible although I can't really imagine anyone saying them. I suppose if I worked at a carwash, I might need to tell a colleague that the next car on the list to be washed has arrived but it's a convoluted context.
Let's go on to your reductions:
1. The car to be washed is here.
It's grammatically correct but again I can't think of a natural context for it.
2. The car to wash is here.
That's questionable. If someone said that to me, I'd probably wonder if they weren't a native English speaker.
What exactly are you trying to practise here?