Наш семинар начнётся двадцатого сентября, в четверг вечером.

Because начнётся is the future tense of the perfective verb начаться meaning to begin / to start.

• начаться = perfective, one completed beginning
• начнётся = it will begin
• it is 3rd person singular, because the subject is семинар = seminar

Russian perfective verbs usually form the future with one simple word, so начнётся naturally means will begin.

The subject is семинар, which is singular and masculine.

So the verb has to agree with it:

• семинар начнётся = the seminar will begin
• not a plural form, because there is only one seminar being talked about

The possessive наш is also singular for the same reason.

Наш means our, and it changes form to match the noun it describes.

Since семинар is:

• masculine
• singular
• nominative

the correct form is наш.

Compare:

• наш семинар = our seminar
• наша лекция = our lecture
• наше занятие = our class
• наши курсы = our courses

In Russian, dates are usually expressed with:

• an ordinal number for the day
• the month in the genitive

So:

• двадцатого = of the twentieth / on the twentieth
• сентября = of September

Together, двадцатого сентября means on the twentieth of September.

This is a very normal pattern in Russian:

• первого мая
• десятого января
• двадцатого сентября

Because Russian uses an ordinal number for the day of the month, not a cardinal number.

• двадцать = twenty
• двадцатый = twentieth

In a date, you need the twentieth, not just twenty.
Here it appears in the required case form: двадцатого.

So Russian thinks more like:

• the twentieth of September

not:

• twenty September

Because the month is in the genitive case after the day of the month.

The basic form is:

• сентябрь = September

But in dates it becomes:

• сентября = of September

This happens with all months in this kind of expression:

• января
• февраля
• марта
• сентября

So двадцатого сентября is the standard way to say on the twentieth of September.

Because Russian uses different patterns for different time expressions.

For days of the week, Russian usually uses:

• в + accusative
• в четверг = on Thursday

But for parts of the day, Russian often uses a bare instrumental form:

• утром = in the morning
• днём = in the daytime
• вечером = in the evening
• ночью = at night

So в четверг вечером literally works like:

• on Thursday, in the evening

It is in the accusative case.

With time expressions like days of the week, Russian often uses:

• в понедельник
• в среду
• в четверг

For четверг, the accusative form looks the same as the nominative because it is an inanimate masculine noun.

So:

• nominative: четверг
• accusative: четверг

Same form, different function.

Вечером is the instrumental singular of вечер.

In this use, the instrumental works adverbially and means in the evening.

This is a very common Russian pattern:

• утром = in the morning
• днём = during the day
• вечером = in the evening
• зимой = in winter

So вечером is not just a noun form here; it functions like a time adverb.

The comma separates an additional clarifying time expression.

First the sentence gives the date:

• двадцатого сентября

Then it adds a more specific clarification:

• в четверг вечером

So в четверг вечером narrows down or уточняет the earlier time phrase. Because of that, a comma is natural and standard here.

In other words, the structure is roughly:

• The seminar will begin on the twentieth of September, specifically on Thursday evening.

Yes. In normal Russian writing, ё is often replaced by е.

So both of these may appear:

• начнётся
• начнется

They represent the same word, but начнётся shows the pronunciation more clearly. For learners, the version with ё is often more helpful.