Studiamo la storia del secolo scorso.

Because Italian usually drops subject pronouns when the verb ending already makes the subject clear.

• studiamo = we study / we are studying
• The ending -iamo already tells you the subject is we

So Studiamo la storia del secolo scorso is perfectly natural.

You could say Noi studiamo la storia del secolo scorso, but that would usually add emphasis, contrast, or clarity, for example:

• Noi studiamo la storia del secolo scorso, voi studiate la letteratura.

Yes. Studiamo is the present tense of studiare.

In Italian, the present tense often covers both:

• we study
• we are studying

So depending on context, Studiamo la storia del secolo scorso can mean either a general activity or something happening now.

If you want to stress that the action is happening right now, Italian can also use:

• Stiamo studiando la storia del secolo scorso

That is closer to We are studying the history of the last century.

Because here storia is being used in a specific way: the history of the last century.

Compare:

• Studiamo storia. = We study history.
  
  • Here storia means the school subject or field in general.
• Studiamo la storia del secolo scorso. = We study the history of the last century.
  
  • Here it is a specific body of history, so the definite article la is natural.

So the article helps make the noun more definite and specific.

Del is a contraction of:

• di + il = del

Here it means of the.

So:

• la storia del secolo scorso
• literally: the history of the last century

This kind of contraction is extremely common in Italian:

• di + il = del
• di + lo = dello
• di + la = della
• di + i = dei
• di + gli = degli
• di + le = delle

Because del scorso secolo is not correct.

If the adjective comes after the noun, you get:

• del secolo scorso

If the adjective comes before the noun, the article changes, because scorso begins with s + consonant, which takes lo, not il:

• dello scorso secolo

So these are possible:

• del secolo scorso
• dello scorso secolo

But del secolo scorso is the more common and natural phrasing here.

Here scorso means last or past.

So:

• il secolo scorso = the last century / the past century

It agrees with the noun it describes:

• secolo scorso → masculine singular
• settimana scorsa → feminine singular
• anni scorsi → masculine plural

This is the same word you see in common time expressions such as:

• l’anno scorso = last year
• la settimana scorsa = last week
• il mese scorso = last month

In Italian, adjectives often come after the noun, and with time expressions this is especially common.

So Italian normally says:

• l’anno scorso
• la settimana scorsa
• il mese scorso
• il secolo scorso

This word order is very natural in Italian, even though English usually puts last before the noun.

So English last century becomes Italian secolo scorso.

Not always. It is a relative expression.

It means the century before the current one, from the speaker’s or writer’s point of view.

So:

• if someone says it today, it usually means the 20th century
• in an older text, it could refer to a different century

For modern everyday use, though, most people will understand il secolo scorso as the 1900s / the 20th century.

Yes, in some contexts it could.

Italian studiamo can be:

• we study / we are studying
  or
• let’s study

The exact meaning depends on context and intonation.

For example:

• Studiamo la storia del secolo scorso.
  This could be a simple statement: We study the history of the last century.

• Studiamo la storia del secolo scorso!
  This could sound like a suggestion or encouragement: Let’s study the history of the last century!

So the form is the same, but the situation tells you how to understand it.