In this context "今日は仕事でお客さんに間違えたメールを送ってしまい、少し落ち込みましたが(#1)、上司に気にしないでいい(#2)と言われてなんとか切り替えられました(#3)。", may I ask
(1) Does "が" mean "but" here?
(2) I guess "気にしない" means "don't mind", but what do "で" and "いい" mean in "気にしないでいい"?
(3) I guess "切り替えられました" mean "change" in a general way, but what's changed here ?

查看翻譯

OK. These are difficult questions, but they are good questions. I will try to explain it as best I can.

About (2)
気にしなくていい and 気にしなくてもいい
Yes. They are roughly the same in this context.

ない
You are right, 'ない' is a negative.
でいい= fine, OK
ないでいい= Negative but that's OK.

A: Do you want something to eat?
B: I'll just have a sandwich. = 俺はサンドイッチでいいよ。
C: I don't need to eat anything. = 僕は何も食べないでいい。

About (3)
切り替える= change, switch
切り替えられました
This example is a typical complication in the Japanese language.
There are three possibilities for what is meant by '切り替えられました', determined by the context.
1. polite "he/she changed."
2. (something/someone) was changed. (passive)
3. managed to change, was/were able to change

In other words, just saying '切り替えられた' does not determine whether it is a polite word, passive or active, it can only be determined by the context. Here, it is used in the sense of number 3.

As for (3), it is natural to have doubts. In fact, even native speakers get the grammar of 'られる' wrong or confused, and it takes them years to understand it, so it's impossible to explain it all here. It is recommended to learn patiently and be exposed to many examples of usage.

I hope this has been helpful.
Thank you.