How do I position the player to the right, left, top, bottom? Are there any detailed instructions in order to style the bar? Where is the style sheet?

Please look at the following images once.

Single Mode
Master
Slave
Player Only
Search Form

It is a Japanese article.

Google VideoBar can be made the Player or VideoBar or Search Form by copying it.

I understand that but how do I change the size of the thumbnails to a custom size other than "small and large"?

How do I custom set the number of thumbnails other than "small and medium"?

How do I get the video details under the thumbnails?

How do I style the thumbnails, I don't see a style.css sheet?

How do I position the floating video player?

Is there a list of custom strings that can be used in the "custom style" box?

Please see Google API about details of the setting.
http://www.google.com/uds/solutions/videobar/index.html

how do I change the size of the thumbnails to a custom size other than "small and large"?

No only small or medium.
Because this is a parameter of Google API, it is not possible to change.

How do I custom set the number of thumbnails other than "small and medium"?

No only small or large.
Because this is a parameter of Google API, it is not possible to change.

How do I get the video details under the thumbnails?

It is not possible to do.

How do I style the thumbnails, I don't see a style.css sheet?

Please change with your stylesheet or Custom Style.

How do I position the floating video player?

Please make the module of Player only and arrange it.

Is there a list of custom strings that can be used in the "custom style" box?

It is possible to do also with Style Sheet of your template.