Learn Hardcore Polish: Latem najczęściej noszę lekką koszulkę i krótkie spodnie. - In the summer I most often wear a light T-shirt and short trousers.

Latem is the instrumental form of lato used adverbially, meaning in (the) summer / during summer.
You cannot say w lato; that is ungrammatical.
You can say w lecie (locative: w + lecie) and it also means in summer, but Latem sounds a bit more natural and typical in general statements about habits.

What exactly does najczęściej mean, and how is it related to często and zwykle?

często = often
częściej = more often
najczęściej = most often

Literally, najczęściej means most often, but in everyday speech it can function very close to English usually / most of the time.
zwykle also means usually / normally, but it doesn’t have the “more / most” comparison built into it.

What tense/aspect is noszę, and how does it compare to English I wear vs I am wearing?

Noszę is the 1st person singular present of the imperfective verb nosić.
Polish has only one present tense form, so noszę can correspond to both I wear (habitually) and I am wearing (right now).
In practice, with a time expression like Latem, noszę is clearly habitual: In summer I usually wear…

Why do we say lekką koszulkę, with -ą and -ę at the end?

Koszulka is a feminine noun. As a direct object it takes the accusative: koszulkę (fem. sg. acc. ends in -ę).
The adjective must agree with the noun in gender, number, and case, so lekka (fem. sg. nom.) becomes lekką (fem. sg. acc., ending -ą).
So:

nominative: lekka koszulka (a light T‑shirt – subject)
accusative: lekką koszulkę (a light T‑shirt – object)

Why is it koszulkę and not koszulę? What is the difference between koszulka and koszula?

koszula = a shirt (usually with a collar and buttons, more formal, like a dress shirt).
koszulka = a lighter, more casual “little shirt”, often a T‑shirt, sports jersey, etc.

In this sentence we are talking about casual summer clothes, so koszulkę naturally suggests a T‑shirt or light top, not a formal shirt.

Why do we say krótkie spodnie? What is special about the noun spodnie?

Spodnie (trousers, pants) is always plural in Polish; there is no singular form for one pair.
Grammatically it behaves like a non‑masculine plural noun, so the adjective is also plural: krótkie spodnie (short trousers/shorts).
As a non‑masculine inanimate plural, its accusative form is the same as the nominative:

Te spodnie są krótkie. – These trousers are short.
Noszę krótkie spodnie. – I wear short trousers/shorts.

Does lekką describe both the T‑shirt and the shorts, or only the T‑shirt?

In lekką koszulkę i krótkie spodnie, lekką clearly modifies only koszulkę. The shorts are described only as krótkie (short).
If you want to say that both the T‑shirt and the shorts are light, you would normally repeat the adjective:

noszę lekką koszulkę i lekkie spodnie

(The same thing happens in English: a light T‑shirt and shorts normally implies only the T‑shirt is light.)

Can the word order be different, e.g. Najczęściej latem noszę… or Noszę najczęściej latem…?

Yes, Polish word order is quite flexible. All of these are grammatically correct:

Latem najczęściej noszę lekką koszulkę i krótkie spodnie. (neutral: time → frequency → verb → object)
Najczęściej latem noszę lekką koszulkę i krótkie spodnie. (slight emphasis on “most often”)
Najczęściej noszę lekką koszulkę i krótkie spodnie latem. (puts latem at the end for extra focus)

The basic meaning stays the same; moving latem or najczęściej mainly changes emphasis.

Which grammatical cases appear in this sentence?

Latem – instrumental singular of lato, used adverbially as a time expression.
najczęściej – adverb (no case).
noszę – verb.
lekką – adjective, feminine singular accusative.
koszulkę – noun, feminine singular accusative (direct object).
i – conjunction.
krótkie – adjective, plural accusative.
spodnie – noun, plural accusative (direct object).

Both clothing items are in the accusative because they are direct objects of noszę.

Could I say Latem najczęściej ubieram lekką koszulkę i krótkie spodnie instead of noszę?

That sounds unnatural in Polish. Ubierać is used mainly as:

ubierać się – to get dressed
ubierać kogoś – to dress someone
ubierać choinkę – to decorate a Christmas tree

For clothes as objects, you normally say:

zakładać koszulkę – to put on a T‑shirt
nosić koszulkę – to wear a T‑shirt (habitually)

So the natural version is Latem najczęściej noszę lekką koszulkę i krótkie spodnie.

Why is there no word for a or the in lekką koszulkę?

Polish does not use articles like a/an or the.
The phrase lekką koszulkę can correspond to a light T‑shirt, the light T‑shirt, or simply light T‑shirt, depending on context.
If you need to show that something is specific (like English this / that / the), you use demonstrative pronouns, e.g. tę lekką koszulkę – this/that light T‑shirt.