Hi Ming-Chou

The ticks need to be paired with an index

ytickValues = Style[#, 22, Bold, Black] & /@ data[[All, 1]];
xtickValues = Style[#, 22, Bold, Black] & /@ Rest@Keys@First@data;

ytick = Transpose[{Range@Length@ytickValues, ytickValues}];
xtick = Transpose[{Range@Length@xtickValues, xtickValues}];

MatrixPlot[Values@data[[All, 2 ;;]], FrameTicks -> {{ytick, None}, {None, xtick}}]

Not sure what you mean by

Is it possible to adjust the width of individual cells?

You can use the AspectRatio option to change the relative width and height of the cells.

You can use

Mesh -> All,
MeshStyle -> Directive[Thickness[.005], White],

to add a uniform spacing between cells.

Also not sure what you mean by

For example, the label string length in the last line is greater than the other line

Do you mean that y4 is wider than y1? If so, you could try using a fixed-width font for the labels.