Learn Hardcore Russian: Я ношу свой кошелёк в сумке каждый день.

Носить is the imperfective, multidirectional verb meaning “to carry habitually or regularly.”
Нести is the imperfective, unidirectional verb meaning “to carry (right now) in one direction, from point A to point B.”
In Я ношу свой кошелёк в сумке каждый день you’re describing a daily habit, so you use ношу. If you wanted to say “I am carrying my wallet (at this very moment),” you’d say я несу свой кошелёк.

Why is the possessive pronoun свой used here instead of мой?

Свой is the reflexive possessive pronoun. It always refers back to the subject of the sentence.
Because я (the subject) owns the wallet, you use свой: Я ношу свой кошелёк (“I carry my own wallet”).
Using мой in this context is grammatically possible but stylistically odd and can create ambiguity about who the owner is. Native speakers almost always use свой when the possessor is the subject.

Why is кошелёк spelled with ё, and where does the stress fall?

The letter ё represents the sound /jo/ and is always stressed.
In кошелёк, the stress is on the ё: ко-шe-лёк.
In everyday print Russians often replace ё with е, but this can lead to mispronunciation since е can be unstressed or pronounced /je/.

Why is кошелёк in the accusative case, and why doesn’t its form change?

Кошелёк is the direct object of the verb ношу, so it takes the accusative case.
Because кошелёк is an inanimate masculine noun of the second declension, its accusative form is identical to its nominative form: кошелёк.

Why is в сумке in the prepositional case? How would it change if I meant “into the bag”?

В + prepositional indicates location (“in the bag”) → в сумке.
If you want to express motion “into the bag,” you’d use в + accusative → в сумку.

Why is каждый день in the singular, not каждые дни?

In Russian “every day” is always каждый день: both words are singular, nominative, and agree in gender.
День is masculine, so the adjective каждый is also masculine singular.
Каждые дни is not used in standard Russian.

Can I change the word order in this sentence?

Yes. Russian has fairly free word order. You can place каждый день (or в сумке) at the beginning or move the object around for emphasis. All of these are grammatically correct, with slight shifts in what you emphasize:

Каждый день я ношу свой кошелёк в сумке.
Я каждый день ношу свой кошелёк в сумке.
Я ношу в сумке свой кошелёк каждый день.