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When you add a column to a table it will automatically increase the table by the width of the column. You can then Shift-drag it back with the Type Tool to the edge of the column from whence it sprang and all of the columns will be reduced equally and the original dimension will be retained.
If you copy and paste an existing table into a new frame that is smaller in either dimension than the dimensions of the table it will be treated as if it were overflow text and not appear in the frame until the frame is enlarged. If you want to enlarge the table proportionally you can drag a corner of the outer frame with command/control-shift pressed and the regular selection tool chosen in the tool panel. If you want it sized unproportionally then do it without the shift pressed.
To make enlarged or reduced in width columns enlarge to the left, right or center place the text cursor outside the table and choose left, right or center alignment as if it were a piece of text. Left alignment will enlarge to the right, center out from center and right alignment will enlarge to the left.
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If you hold down the Shift key as you drag the right-most edge of a table, it'll resize the whole table horizontally while keeping the relative widths of the columns the same. I've found that helps a great deal when trying to get a table and a text frame to "kiss fit."
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For your question 1.
I'm wondering if it's possible to add new columns to a table without changing the width of a table. So that the other columns will become smaller.
I agree to what Bill said above.
For question 2.
Also, is it possible to automatically fit an existing table into a new frame with different dimensions?
Then No. However, you can try setting up auto-size properties on the new frame first and then try copy pasting the table.
For question 3.
how to change the width of a column so that the table resizes to one side only? Currently table resizes to both sides 1/2 way of the amount the column was resized.
Depends on how you have aligned the table. To try the same, do this...
1. Enlarge the frame containing the table.
2. Place the cursor just outside the table so that it is of the same height as the table
3. Click on right align from the control strip so that you get something like...
4. Now when you increase the table size, it will grow towards the left.
You can see similar (opposite) effects when the table is aligned to the left. When the table is aligned centrally, it will grow half and half on both left and right.
There is a script that fits a table to the width of the text frame, if that's what you're needing. Let me know and I can try to remember the script name!