Mary Queen of Scots was a Stuart, not a Tudor. Elizabeth had her executed because her claim to the throne was arguably stronger than Elizabeth’s own.
Yes.
There is irony in Elizabeth's eventual successor. I'm not sure this is true, but I've heard that when Elizabeth was old and dying, she lost the power of speech. Childless, she had no obvious heir. Ministers gathered around her deathbed as she lay dying and tried to determine the successor. Various names were mentioned. Each had pros and cons. Finally, they mentioned James Stuart, ruling Scotland as James VI. When they mentioned him, Elizabeth managed to raise her hands to her head, fingers splayed upward, making a crown. And that's how James VI of Scotland became James I of England, and how the Stuarts made it to the throne.
But the story is probably not true, I think, because Elizabeth likely could have just written a note.
As Henry VIII's only survivign child, no one had a stronger claim than Elizabeth. The only way Mary's (or any Stuart's) claim could be stronger was if one considered Elizabeth illegitimate. Catholics did consider Elizabeth illegitimate because they did not recognize Henry's divorce and remarriage. But by the laws of the time, after Henry had "protestantized" the country, she was legitimate.