Zu Beginn des Seminars legt die Dozentin ein Handout auf jeden Tisch.

Because German main clauses usually follow the verb-second rule. The first position is taken by Zu Beginn des Seminars, so the finite verb legt must come next.

Compare:

• Die Dozentin legt ein Handout auf jeden Tisch.
• Zu Beginn des Seminars legt die Dozentin ein Handout auf jeden Tisch.

In both cases, the verb is still in second position.

It is a time expression meaning at the beginning of the seminar.

• zu Beginn = at the beginning
• des Seminars = of the seminar

This whole phrase sets the time for the action and is placed first in the sentence for emphasis or flow.

Because Beginn is followed here by the genitive, which often expresses of in English.

So:

• das Seminar = nominative
• des Seminars = genitive, meaning of the seminar

This is a very common pattern in more formal German:

• zu Beginn des Kurses
• am Ende des Tages
• während des Semesters

Yes, largely. Zu Beginn is a common and natural expression meaning at the beginning of. It is especially common in formal or written German.

A more everyday alternative is often am Anfang:

• Zu Beginn des Seminars = a bit more formal
• Am Anfang des Seminars = very common, slightly more conversational

Both are correct in many contexts.

The ending -in usually makes a noun feminine when talking about a female person.

• der Dozent = male lecturer/instructor
• die Dozentin = female lecturer/instructor

This pattern is very common:

• der Lehrer → die Lehrerin
• der Student → die Studentin

Because Handout is neuter in German: das Handout.

In the accusative:

• masculine: einen
• neuter: ein
• feminine: eine

So:

• ein Handout is correct
• einen Handout would be wrong

Because Tisch is masculine, and here it is in the accusative singular.

The word jeder changes by case and gender:

• nominative masculine: jeder Tisch
• accusative masculine: jeden Tisch

So jeden is the correct form because the sentence uses the accusative.

Because auf is a two-way preposition. It can take:

• accusative when there is movement toward a destination
• dative when there is location with no movement

Here, the handout is being put onto the table, so German uses the accusative:

• Sie legt ein Handout auf jeden Tisch.

If you were describing where something already is, you would use the dative:

• Das Handout liegt auf jedem Tisch.

Because jeder/jede/jedes usually works like each or every, and German normally uses a singular noun after it.

So:

• jeden Tisch = each table / every table

This gives the idea of one table at a time, distributively.

You could also say:

• auf alle Tische

But that means onto all the tables, not quite the same structure as each table.

Legen means to lay or to put something down, usually with the idea that the object ends up lying flat.

That fits a Handout very well, since a handout is normally placed flat on a table.

German often distinguishes these verbs more clearly than English:

• legen = lay something down
• stellen = stand something upright
• setzen = set someone/something into a sitting position

So legen is the natural choice here.

This is a very important German pair:

• legen = to lay / to put down something
  
  • action
  • usually takes a direct object
• liegen = to lie
  
  • state/location
  • no direct object

Examples:

• Die Dozentin legt ein Handout auf den Tisch.
• Das Handout liegt auf dem Tisch.

So in your sentence, legt is used because the lecturer is actively placing the handout.

Yes. It is a very common loanword from English in German, especially in school, university, and business contexts.

Typical forms are:

• singular: das Handout
• plural: die Handouts

German also has other words depending on context, such as Unterlagen or Skript, but Handout is completely normal.

Because in German, all nouns are capitalized.

In this sentence, the capitalized nouns are:

• Beginn
• Seminars
• Dozentin
• Handout
• Tisch

This is one of the most noticeable differences from English spelling.