In To Kill a Mockingbird, what is Harper Lee's tone toward Tom Robinson's death? Does she want us to believe that he was shot while escaping?

The author's tone is defined as the attitude behind the subject matter--in this case, Tom Robinson's death. Harper Lee chose to have Atticus break the news of Tom's death to other characters, which means that the news comes by way of good authority. For example, Scout doesn't learn about Tom's death through Stephanie Crawford, the town gossip; she hears it from Atticus, a man who is respected by the community and wouldn't lie or joke about a subject so serious. Therefore, one could infer that the tone behind the news of Tom's death is critical and instrumental to the story and its characters. Lee also chooses to have Atticus simply blurt out, "He's dead," in Chapter 24, rather than to have him use a euphemism to soften the blow. Atticus is upset that Tom is dead. He was very determined to return Tom to his family after going through an appeals process. Not only that, but Tom was illegally tried and a victim of a racist society; so, that makes Atticus frustrated as well. Hence, the tone regarding Tom's death is frustration and disappointment as shown through the character of Atticus. 

In response to the second question, yes, Tom died trying to escape. While he was in prison, he charged for the fences during exercise time and almost made his way to freedom before he was shot to death. The guards reported that if he had had two good arms, he would have escaped completely. As it was, they shot up a few warning shots, but Tom kept running. In the end, he was shot multiple times and died there.