You need to copy the content of the cell not the cell itself. This takes a bit of practice and is mightily annoying!
If you see the grey, you've selected too much. The easiest way is to click in the cell and press ctrl-a then copy. If the paste doesn't do what you expect, check what you've set the default paste to. You can, of course, control the paste type after you paste or you can do a paste special.
UPDATE: Oops! Never noticed that ctrl-a selects everything and I've been using Word for decades!
As an alternative. Click at the start of the cell, then shift-end. Then still holding down shift, use the back arrow to go back one. That last part is important as it changes from selecting the cell to selecting the content. I actually tried it this time so hopefully no more embarrassments.
UPDATE 2: Now that we know you want to copy the format and not the content, we can focus on the best way to do that. Which is to create a style and apply the style to all the cells you want to have the same style. You can also assign a keyboard shortcut to a style to make things even easier.
UPDATE 3: Just to be absolutely clear. The format for a paragraph is contained in the (normally hidden) end paragraph mark. It is this that is selected when you do shift-end, turn on visible paragraph marks to see this more clearly. In a table, this changes to a cell marker rather than a paragraph on. To copy the formatting for a paragraph (or cell), you have to include that mark in your selection.
To Turn a formatted cell into a style, show the style panel, select the cell, click on the "New Style" button. The resulting new style dialogue should already be formatted as per the cell. Give this a name and set the shortcut. Save the style to normal.dot if you want it always available otherwise save it to the document. Now in any cell, simply select your style (no need to select the cell) and the cell will be formatted accordingly.
The "extra thought" is a one-off and need never be repeated if you save the style in normal.dot (the default Word template).