What is the small solid black square Show/Hide symbol in my table row?

The small black square indicates that the paragraph inside the cell has one of these format items turned on in the Line & Page Breaks tab of the Paragraph dialog: "Keep with next", "Keep lines together", or "Page break before". This is not limited to paragraphs in table cells -- it will appear to the left of any paragraph that has one of those items checked.

I very much doubt that any of these are related to not being able to change the row height. Something else is going on in that table, and it isn't anything ordinary or intentional. It may be some damage or corruption in the internal description of the table.

There are several ways to try to wipe out table corruption. Probably the easiest is to go to the Table Tools > Layout tab of the ribbon and click the Convert To Text button; then, while the text is still selected, go to the Insert tab, click the Table button, and choose Convert Text to Table. Then try changing the row heights.

If that doesn't help, another method is to create a new table and copy/paste the cells (avoid selecting the end-of-row markers outside the right end of each row) from the old table to the new one.

Using the Replace dialog, put this expression in the Find What box:

.^#^#

and put this expression in the Replace With box:

^& /m

Select the column (or any part of the table that you want to change) and click the Replace All button.

The Find What expression consists of the decimal point and two copies of the code ^# for "any digit"; this will match the decimal point and the following two digits in each cell. (The amounts to the left of the decimal points in the table can have any number of digits.)

The Replace With expression consists of the code ^& for "the matched text" and the /m that you want to add after the digits.