Après le cours, elle ramasse chaque questionnaire et note la présence des élèves.

In French, un cours can mean:

• a class session
• a lesson
• a course, depending on context

In Après le cours, the most natural meaning is After class or After the lesson, because the sentence describes something that happens right after a teaching session ends.

So cours is a broader word than English course.

Après is followed directly by a noun phrase, and here French uses the definite article le:

• après le cours = after the class

You would not say après du cours here.
Du usually means some or is part of other structures, but after après in this kind of sentence, French normally uses the regular article for the thing being referred to.

Ramasser literally means things like:

• to pick up
• to gather
• to collect

In this sentence, elle ramasse chaque questionnaire means she is collecting the questionnaires after class.

So even though ramasser can sometimes suggest physically picking something up from the ground, here it means to gather in / collect.

Because ramasse is the conjugated verb, while ramasser is the infinitive.

Here the subject is elle, so in the present tense:

• je ramasse
• tu ramasses
• il / elle ramasse

So elle ramasse means she collects.

The same thing happens with note:

• elle note = she notes / records

Because chaque is followed by a singular noun in French.

• chaque questionnaire = each questionnaire
• chaque élève = each student

Even though the meaning refers to multiple items one by one, the noun stays singular after chaque.

This is different from tous les questionnaires, which means all the questionnaires and uses a plural noun.

Both can refer to a whole set, but they do it differently:

• chaque questionnaire = each questionnaire, considered individually
• tous les questionnaires = all the questionnaires, considered as a group

In your sentence, chaque questionnaire emphasizes that she collects them one by one.

French often uses a noun expression where English prefers a verb expression.

Here:

• noter la présence literally means to note/record the presence
• in natural English, that often becomes to take attendance or to mark who is present

So this is a normal French way to express the idea of attendance.

French often uses the definite article where English may not.

So:

• la présence des élèves literally = the presence of the students
• natural English = attendance or the students’ attendance

This article is part of the normal French noun phrase.
French generally likes to specify nouns with an article, especially in expressions like this.

Here des élèves means of the students.

The structure is:

• la présence de + noun
• la présence des élèves = the presence of the students

This is de + les = des.

So this des is not the some form. It is the contraction of:

• de + les → des

Because both verbs have the same subject.

• elle ramasse ... et note ...

French, like English, usually does not repeat the subject when it stays the same:

• She collects each questionnaire and records attendance.

You could repeat elle, but it would usually sound unnecessary here.

Élèves can refer to:

• male students
• female students
• a mixed group of students

The singular forms are:

• un élève = a male student
• une élève = a female student

In the plural, des élèves can refer to any group of students. The spelling stays the same in the plural except for the article and the plural s.

Yes, depending on context.

Un questionnaire can mean:

• questionnaire
• survey
• form

In a classroom context, questionnaire often suggests a worksheet, quiz sheet, survey, or set of written questions.
So elle ramasse chaque questionnaire could mean she is collecting each student’s questionnaire sheet.