Laitan takuukortin laatikkoon ja luen käyttöohjeen rauhassa kotona.

That -n is the common “total object” marking (often called accusative, but it looks like the genitive in the singular). It typically appears when the action is seen as complete/whole:

• Laitan takuukortin… = I put the warranty card (as a whole)…
• Luen käyttöohjeen… = I read the instructions (as a whole, i.e., read through them)

Yes. Using the partitive would usually make the action incomplete/ongoing/partial:

• Laitan takuukorttia laatikkoon would sound odd in most normal contexts (because putting something into a box is usually a complete event), but it could work in special contexts (e.g., fiddling/trying to fit it in, or focusing on the process).
• Luen käyttöohjetta rauhassa kotona = I read the manual some / for a while / not necessarily all of it.

Laatikkoon is the illative case, which answers “into where?” and expresses movement into something.

• laatikko = a box
• laatikkoon = into the box

The illative ending here is -oon, giving laatikko + -on → laatikkoon (with a long oo in writing).

The dictionary form is laittaa (to put). When you conjugate it, Finnish applies consonant gradation:

• laittaa (strong grade tt)
• laitan (weak grade t) = I put / I will put

So tt → t in this form.

Because lukea (to read) has a stem change in the present tense:

• lukea → present stem lue-
• luen = I read / I will read

So it’s a common irregular-ish pattern where ke changes and you get lue- in the present.

Rauhassa is the inessive case of rauha (peace): rauha → rauhassa = in peace. Finnish often uses this structure adverbially:

• lukea rauhassa = to read peacefully / calmly / without rush

Kotona is the essive case form used in the fixed locational sense at home:

• koti = home
• kotona = at home

Even though Finnish has many location cases, kotona is the normal, idiomatic way to say at home (state/location).

Usually it suggests a natural sequence (first put the warranty card in the box, then read the manual), but it doesn’t strictly encode timing. If you want to emphasize “and then,” you could add something like sitten:

• Laitan … ja sitten luen … = I put … and then I read …

Yes. Finnish word order is flexible, and changes usually affect emphasis rather than basic meaning. For example:

• Kotona luen käyttöohjeen rauhassa. emphasizes at home.
• Käyttöohjeen luen rauhassa kotona. emphasizes the manual (as opposed to something else). The neutral, straightforward order is like your original sentence.