Learn Hardcore Finnish: Tämä sääntö auttaa meitä pysymään turvassa.

Tämä means this (singular), and it agrees in number with the noun.
sääntö (rule) is singular here, so you use tämä sääntö = this rule.
Tämän säännön would be the genitive of this rule (used for possession, objects of some verbs, etc.), not the subject form.

What grammatical role does sääntö have here, and what case is it in?

sääntö is the subject: it’s the thing doing the helping.
It’s in the nominative case (the basic dictionary form), which is the normal case for subjects in affirmative sentences.

How is auttaa formed here? What tense/person is it?

auttaa is the verb to help.
It’s in the present tense, 3rd person singular: (it) helps.
Finnish doesn’t need an explicit subject pronoun like it here; the subject is already Tämä sääntö.

Why is it meitä and not meidät?

meitä is the partitive plural of me (we/us).
The verb auttaa very commonly takes its object in the partitive: auttaa jotakuta = help someone.
meidät would be accusative (often implying a completed/total result), and it generally doesn’t fit the normal pattern with auttaa in this meaning.

Is meitä an object, even though English would say “help us”?

Yes.

meitä is the object of auttaa.
Finnish marks objects with cases rather than word order, so meitä is clearly “the people being helped” because it’s the object form required here.

What is pysymään exactly? Why does it end in -mään?

pysymään comes from the verb pysyä = to stay / to remain.
It is the MA-infinitive illative form (often taught as “the -maan/-mään form”).
This form is used after certain verbs (including auttaa) to express “help (someone) to do something”:
- auttaa meitä pysymään = helps us to stay

How do I know it’s -mään and not -maan?

It’s vowel harmony.

Finnish uses front vowels (ä, ö, y) vs back vowels (a, o, u).
Since pysyä contains front vowels (y, ä), the ending is -mään (front-vowel version), not -maan.

What case is turvassa, and why is it used?

turvassa is the inessive case (-ssa/-ssä), meaning roughly in.
turva relates to safety/protection, and turvassa is a common expression meaning safe / in safety.
So pysymään turvassa is literally like to stay in safety → natural English: to stay safe.

Could I also say turvallisena or something like that instead of turvassa?

Sometimes, but it changes the feel.

turvassa is the most idiomatic for “safe” in the sense of being in a safe situation/place.
turvallisena (essive: “as safe”) can be used in some contexts, but it often sounds more descriptive/abstract and is less “set phrase” than turvassa.
For this sentence, pysyä turvassa is the natural choice.

Can the word order change? For example, could I say Tämä sääntö auttaa pysymään turvassa?

Yes, with a small change in emphasis/explicitness.

Tämä sääntö auttaa meitä pysymään turvassa. = explicitly helps us stay safe.
Tämä sääntö auttaa pysymään turvassa. = more general: This rule helps (one/people) stay safe, with the “who” left implicit.
Finnish word order is flexible, but leaving out meitä removes that explicit object.