I can't find anything in past episodes that indicates that Bates ever knew Baxter. No, I don't think there is any link there. I think there is some type of family relationship between Thomas and Baxter, and that whatever the deeper (deeper than the theft itself, but the "what became of the stolen items" bit) hidden information is, Baxter cannot betray Thomas re the matter because it would hurt someone Baxter loves. A child, a brother or sister, a parent. SOMEone.
It seems fairly obvious that Baxter has a very strong desire to "go straight" and have a decent life. But say what happened to the jewels, that she will not. No, she is protecting someone here; and that someone is not herself.
I don't like the schoolteacher either, Steph. I agree with a lot of her political views, but she is one of those strident, bully types I abhor. Obviously, she has some good attributes, but she is poison to Tom. I want Tom and Sibby to stay at Downton Abbey. I guess basically because he loved Sybil so much and because that is her little girl.