Летом я обычно ношу майку, шорты и лёгкую кепку.

Летом is the instrumental singular form of лето, and Russian often uses this form to mean in summer / during the summer.

This is a very common pattern with seasons and some time expressions:

• летом — in summer
• зимой — in winter
• утром — in the morning
• вечером — in the evening

So Летом я обычно ношу... is the natural way to say this.
В лето is not normal in this meaning.

Yes. Russian often drops subject pronouns when they are clear from the verb form.

Ношу already tells you the subject is I because it is 1st person singular. So all of these are possible:

• Летом я обычно ношу...
• Летом обычно ношу...

Including я sounds perfectly natural too. It can make the sentence a bit clearer or slightly more emphatic.

Обычно means usually. Its position is fairly flexible.

These are all natural:

• Летом я обычно ношу...
• Я обычно летом ношу...
• Обычно летом я ношу...

The basic meaning stays the same, but the emphasis shifts a little depending on word order. The version in your sentence is very neutral and natural.

Because носить means to wear in the sense of wearing something regularly / as a habit / as typical clothing.

• носить — to wear
• надевать / надеть — to put on

So:

• Я ношу майку. — I wear a tank top / undershirt.
• Я надеваю майку. — I put on a tank top / undershirt.

In your sentence, the idea is about what the speaker typically wears in summer, so ношу is the right choice.

They are in the accusative case because they are direct objects of ношу.

Base forms:

• майка
• кепка
• лёгкая кепка

Accusative forms:

• майку
• кепку
• лёгкую кепку

For feminine singular nouns like майка and кепка, the accusative commonly changes -а to -у.
The adjective also changes to match the noun:

• лёгкая → лёгкую

Шорты is a plural-only noun in Russian, like shorts in English.

Also, for inanimate plural nouns, the accusative is the same as the nominative. So:

• nominative: шорты
• accusative: шорты

That is why it stays the same in this sentence.

Майка usually means a sleeveless shirt, tank top, or undershirt.

It is not usually the normal word for T-shirt. For T-shirt, Russian normally uses футболка.

So this sentence suggests something lighter and more sleeveless than a regular T-shirt.

Кепка is a cap, usually one with a visor or brim, such as a casual cap or baseball-cap-type hat.

It is different from:

• шапка — a hat/cap, often soft or warm, especially for cold weather
• шляпа — a hat, often more formal or with a full brim

So кепка fits very naturally with summer clothing.

Yes, ё is important.

In лёгкую, the stress is on лё:

• лЁгкую

A rough English approximation is LYOHG-ku-yu, though the real Russian pronunciation is softer and quicker than that.

A useful thing to remember: ё is always stressed.
In many printed texts, Russians sometimes write е instead of ё, but the correct word here is still лёгкую.

No. Russian word order is quite flexible.

Your sentence:

• Летом я обычно ношу майку, шорты и лёгкую кепку.

is a very neutral, natural order.

But Russian could also say:

• Я летом обычно ношу майку, шорты и лёгкую кепку.
• Обычно летом я ношу майку, шорты и лёгкую кепку.

The main difference is emphasis and focus, not core meaning.

Because this is a simple list:

• майку, шорты и лёгкую кепку

In Russian, just like in standard English list punctuation, you normally put commas between earlier items, but not before the final и in a simple list.

So this punctuation is completely normal.