… I'd like to write without using any time signature at all.  This means that my measures can have differing numbers of beats.  Can I do this in Finale? …

Yes, this is possible in Finale.

Each measure must have a time signature (because of the way Finale works), but the time signature does not have to display.

… What if I want ten beats to a measure, and the time signature tool does not have that option? …

But it does have that option.

As a matter of fact the Time Signature Tool is a very powerful tool.

You can have any number of beats - with any duration.

Furthermore, you can use invisible barlines. Any barline can be invisible.

This can be done “horizontally” through through the measures in a certain staff (use the Staff Tool).

And it can be done “vertically” through all the staves in a certain measure (use the Measure Tool to select Invisible Barline).

It is possible but there might be some small spacing issues.

No trouble with any of that, and I do have to move notes around to make them look right. However, after inputting notes and markings, I decided I wanted to add a few extra notes.  I sure can't figure out how to do that without re-doing the whole thing! Any ideas appreciated.

If you do it that way, you will have spacing issues. If you are going to use various meters in every measure can you not use several huge measures like 100/4?

Thanks for verifying my conclusion upon waking up in the middle of the night (which I often do when I have a deadline). I still can't figure out how to edit the measure I've already input. I guess make a long measure, but not too long, whose time signature only goes until the start of the next measure?  I tried that once and notes got piled up in the left side and were pretty hard to move around correctly.

If you specify a huge meter and fill it up there should be no problem with spacing or very little. Make sure auto update is on. You can also hide bar lines or use invisible bar lines.
Using a non standard notation like that is always going to need some adjustments.