What is the one word for someone who gets worried and anxious too fast, usually over silly things?

Solution 1:

This is a noun, but such a person is a worrywart. It is frequently used to describe loved ones who are overly concerned about your well being. It fits perfectly into your example sentences.

My mother is a worrywart.

I want to tell my mother about my flight tonight, but, you know, she is a worrywart and will lose sleep worrying over my safety.

This word is very informal, but not derisive. It typically implies some measure of affection.

The expression is used mainly in American English.

Worry wart:

1956, from comic strip "Out Our Way" by U.S. cartoonist J.R. Williams (1888-1957). According to those familiar with the strip, Worry Wart was the name of a character who caused others to worry, which is the inverse of the current colloquial meaning.

(Etymonline)

Solution 2:

This state of mind has a rich tradition in Victorian times, where upper-class females had little to spend their time on. An entirely appropriate term for it would be high-strung.

Having an extremely nervous or sensitive temperament.
"I'm not blaming my kid, but she's very high-strung."

Reference:
https://www.merriam-webster.com/dictionary/high-strung