TILED IMAGES CAPTION

I would like to know if it is possible to make the caption on tiled galleries always shown and not hide when mouse is not over.

Jetpack doesn’t include such an option, but you should be able to always display the captions with some Javascript targeting .tiled-gallery-caption.

how could I change text color and background color on captions?

You can do so with some custom CSS. Try for example adding the following CSS to your theme stylesheet or your custom CSS editor, available under Appearance > Edit CSS:

.tiled-gallery-item .tiled-gallery-caption {
	background: red;
}