Are typing skills essential for writing GRE? What kind of word-processing tool is used by ETS in GRE?

Typing involved in GRE tests is bare minimum. In fact, except for the Analytical writing assessment (AWA) and Numeric entry in quantitative reasoning section, there is no real typing involved. Since you have write two essays in the AWA which involves writing for an hour some type skills are desirable though. If you can type minimum 25-30 words per minute that will be enough for you to see through the test. People with better typing skill will have some advantage loaded in favor of them.

The word processing application used by ETS is very primitive though it is kept deliberately in that way to ensure that the people with better understanding of the language are benefitted. The basic functions such as cut and paste, undo, delete text, and insert text are available in the examination however advanced functions such as spelling or grammar Checking are absent.

So it would be prudent on your part if you work  on your typing a bit. Working on your spellings and grammar will also help though GRE does not test your skills in these areas directly.