Po przeczytaniu maila od teściowej od razu zadzwoniłam do teścia.

This is a very common Polish pattern:

• po
  • verbal noun in the locative
• it means after doing something

Here:

• przeczytać = to read / to read through
• przeczytanie = the reading / reading
• po przeczytaniu = after reading

So Po przeczytaniu maila... literally works like After the reading of the email..., but in natural English we say After reading the email...

This structure is very common in Polish:

• po przyjściu do domu = after coming home
• po obejrzeniu filmu = after watching the film
• po skończeniu pracy = after finishing work

Because maila is in the genitive singular.

That happens here because przeczytaniu comes from the verbal noun przeczytanie, and verbal nouns in Polish often take their object in the genitive:

• przeczytać mail = to read an email
• przeczytanie maila = reading an email
• po przeczytaniu maila = after reading an email

So the case changes because the verb has been turned into a noun-like form.

Also, mail is a borrowed word, and in everyday Polish maila is a very normal form. You may also see e-maila.

Because od means from, and od requires the genitive.

• teściowa = mother-in-law
• genitive singular: teściowej

So:

• mail od teściowej = an email from my mother-in-law

This is a standard pattern:

• list od kolegi = a letter from a friend
• telefon od szefa = a phone call from the boss
• wiadomość od mamy = a message from mum

Because in Polish you call to someone:

• dzwonić do kogoś
• zadzwonić do kogoś

So the person receiving the call is introduced by do, which takes the genitive.

Here:

• teść = father-in-law
• genitive singular: teścia

So:

• zadzwoniłam do teścia = I called my father-in-law

This is different from English, where we simply say call someone. In Polish, with this verb, you normally need do.

It tells us two things:

1. it is past tense
2. the speaker is female

The ending -łam is the feminine 1st person singular past ending.

Compare:

• zadzwoniłam = I called (female speaker)
• zadzwoniłem = I called (male speaker)

So just from the verb, we know the speaker is a woman.

Because Polish often drops subject pronouns when they are clear from the verb form.

Here, zadzwoniłam already means I called. The I is built into the verb ending, so ja is unnecessary.

You could say:

• Ja zadzwoniłam do teścia

but that would usually add emphasis, like:

• I called my father-in-law

In neutral sentences, Polish usually leaves ja out.

Od razu means immediately, right away, or straight away.

In this sentence:

• od razu zadzwoniłam = I called right away

It shows that the call happened very soon after reading the email.

You will hear od razu all the time in Polish:

• Zrób to od razu. = Do it right away.
• Od razu to zauważyłem. = I noticed it immediately.

Yes, Polish word order is fairly flexible.

The given sentence is very natural:

• Po przeczytaniu maila od teściowej od razu zadzwoniłam do teścia.

But you could also hear:

• Od razu po przeczytaniu maila od teściowej zadzwoniłam do teścia.
• Do teścia zadzwoniłam od razu po przeczytaniu maila od teściowej.

The basic meaning stays the same, but the focus changes a little.

The original version sounds smooth and neutral:

• first the time/background: after reading the email
• then the quick reaction: I immediately called
• then the target: my father-in-law

Because zadzwonić is the perfective verb, and here Polish is describing a single completed action.

• dzwonić = to be calling / to call repeatedly / to call in general
• zadzwonić = to make one call, to call once, successfully as a complete event

So:

• zadzwoniłam do teścia = I called my father-in-law
• dzwoniłam do teścia could mean something more like I was calling my father-in-law, I called him (imperfective context), or sometimes I tried calling

In this sentence, the speaker is telling us about one clear next action after reading the email, so zadzwoniłam fits best.

Yes, in normal modern Polish:

• teściowa = mother-in-law
• teść = father-in-law

They do not tell you by themselves whether these are the spouse’s parents from the husband’s side or the wife’s side; the meaning is just parent-in-law.

So in this sentence:

• mail od teściowej = an email from the mother-in-law
• do teścia = to the father-in-law

Yes: it is a borrowed, everyday word.

In modern Polish, people very commonly say:

• mail
• maila
• mailu

You may also see the more formal or fuller spelling:

• e-mail
• e-maila

In ordinary speech, mail is extremely common and natural. So this sentence sounds modern and normal.