Are the anodes and cathodes in a vacuum tube truly charged?
Are the anodes and cathodes in a vacuum tube truly charged?

They are if you apply a voltage between them.

Can I treat them like any other charged object?

That depends on what the point is, but in the general case, no.  That's because there is a lot more going on in a vacuum tube than just the capacitance between the anode and cathode.

how can I calculate the charge?

The datasheet should give you some idea of the anode/cathode capacitance, or you could measure it.  From that you can find the charge as a function of applied voltage like you would with any capacitor.

However, this isn't a very useful parameter when designing circuits with tubes.

How would I calculate the alternating charge if it's ac?

This question seems to indicate you don't understand how tubes work in the first place.  Tubes only conduct in one direction, so AC isn't relevant, depending on what exactly you mean by that.

In general, what you are asking aren't the kinds of questions one normally asks when designing circuits with tubes.  You should pop up a few levels and explain what you are trying to accomplish, and why you want to use tubes to accomplish it.  Some of them new-fangled transistorators might be just what you need.