Word 2010, tables getting cut off and can't change "keep with next" paragraph option

Unless you really need to have the table wrapped, setting the wrap to None will solve the problem. If you do for some reason need to have it wrapped, then you will need to make sure that a Compatibility Option is set correctly. In File | Options | Advanced: Layout options, make sure that the box for "Don't break wrapped tables across pages" is not checked.

The filled boxes indicate that the settings in the selection are mixed. You'll see this display when a large amount of text is selected even if the settings are uniform across the entire selection because Word just refuses to be specific beyond a certain amount of text.

Note that none of the Line and Page Breaks settings have any effect within a cell or row if you have allowed rows to break across pages. The only setting that has any use in a table is "Keep with next," which is effective only between rows. You generally want your header row to be formatted as KWN; if a table is short enough to fit on one page and you don't want it to split, then apply KWN to all rows except the last. Otherwise, use it only for rows that need to be kept together even if the table is split.

Note that, even if all the rows are set to KWN, any row that has more than one line of text can be broken if you allow rows to break. If all the rows are formatted as KWN and none has more than one line of text, then you'd have a problem even in an inline table (though Word would probably break it anyway--that's what it does with ordinary paragraph text).