With which hand did Mr. Ewell write, and what was Atticus's point in having him write his signature?

Atticus had Mr. Ewell write his name to show that he was left-handed.

The girl was beaten by someone who lead with his/her left hand.

Tom Robinson could not even use his left hand, therefore he could not have beaten her; just proving further that the trail was a farce. Atticus was trying to suggest that it was Mr. Ewell who beat Mayella, not Tom. Mr. Ewell was left-handed, and Tom couldn't even use his left-hand. Atticus then proceeds to question Mayella about her home life*, further driving his point home.