で in 気にしないでください

[Yuta73]

Of all the usages of で, none seems to fit well. Is there something I'm not catching or my grammar book is not covering this case?.
I've got other cases also. I don't have to say they come from a girl.
ただの風邪でよかったわね
あとは、ゆっくり休んでね

See below the explanations from Timwerx:
Action indicator de
Particle de is a preposition that shows us where an action takes place:
Kyou ie de taberu. (I'll eat at home today.)
Kodomotachi wa kouen de asonde imasu. (The kids are playing in the park.)
De is used for "among":
Watashi no yuujin de, piano o hikeru hito ga inai. (There is no one among my friends that can play the piano.)
De also indicates a method:
Onamae wa pen de kaite kudasai. (Please write your name with a pen.)
Genkin de haraimashou. (Let's pay with cash.)
Eigo de hanashite kureru? (Would you please speak English?)
De is sometimes used before ii to say that something is good or sufficient as it is:
Kore de ii. (This is okay. [It's good enough.])
Ashita de ii. (Tomorrow will be okay.)
De is sometimes combined with wa to show that something is done within the subject:
Tokushima dewa maitoshi yuumei na matsuri ga okonawareru. (A famous festival is held in Tokushima every year.)

 

[Half-n-Half]

IIn your first sentence 気にしないでください ～で is used as the negative command. して=do しないで=don't do. Or, negative plain form of the verb plus で. So your sentence says "Please don't worry (about it)."

I'm not sure what you would consider the second one to be, but it basically means "I'm glad it was just a simple cold." The third one is the command form for verbs ending in む、ぬ、ぶ. So 休む becomes 休んで i.e. "Take it easy later, OK?"

Hopefully Toritoribe, Glenn or Elizabeth come along and correct me

EDIT: I suppose the second one is considered the "de ii" construct.

 

[Toritoribe]

As HnH-san already explained accurately, the first and third one is the command/request usage of the -te form, and the second one is the past form of でいい; as in "De is sometimes used before ii to say that something is good or sufficient as it is" in your quote.

As a simple command:

食べて (Tabete): "Eat."
読んで (Yonde): "Read."​

In requests with くれ kure and 下さい kudasai.

本を読んでください: "Please read the book."​

Japanese verb conjugation - Wikipedia

Incidentally, ない has two different -te forms. You need to pay attention which one is appropriate for the construction.
e.g.
食べないでいる
食べないでおく
食べなくてはいけない
食べなくてもいい
食べなくてもかまわない
食べないでほしい
食べなくてすいません

The ない nai ending conjugates in two ways.

1. As an i adjective. For example the past tense of 食べない tabenai is 食べなかった tabenakatta and the te form is 食べなくて tabenakute.
2. There is a special te form made by adding で de. For example, 食べないで tabenaide. This is used, for example, in 食べないで下さい tabenaide kudasai: "Please don't eat (this)".

Japanese verb conjugation - Wikipedia

 

[Yuta73]

I understood everthing now. It's just that I forgot that the ーて form ending becomes ーで under certain conditions. Sorry, newbie thing.

The second one, if I understand well, is:
でいい plain
でよく polite
でよかった polite & past

 

[Toritoribe]

いい(plain)
いいです(polite)
よかった(past)
よかったです(polite past)