Journalists do sometimes make mistakes, but in this case, the sentence is perfectly fine. It's in present tense, not past tense.
1) In this case, "when asked to" means "when she is asked to." The word "asked" is a passive participle here. Here are some examples with a similar structure:
"Ice will melt if left in the sun." (Ice will melt if [it is] left in the sun.)
"If hired, he will be expected to work on weekends." (If [he is] hired...)
The participle describes the noun, in terms of a verb that affects it. You can also say "Ice left in the sun will melt." (What kind of ice? Ice [that is] left in the sun [by someone].) "The candidate hired by our company." The word "hired" describes the person.
2) "Doesn't" is fine. The sentence is in present tense.
3) To "bat an eye" or "bat an eyelid" means to hesitate or show some emotion such as surprise or embarrassment. The phrase is usually used in the negative: we say that a person "didn't bat an eye" in order to show that the person was not surprised, embarrassed, or hesitant. In this case, it shows that the woman is perfectly comfortable telling people what her fee is.
To paraphrase the sentence, you could say: "She tells us her typical fee without hesitation" or "She isn't shy about telling us her typical fee."