Learn Hardcore Russian: В университете скоро будет большая научная конференция. - Soon there will be a big scientific conference at the university.

в + Prepositional case usually means in / inside a place, but with some institutions it corresponds to English at:
- в университете = at the university (literally: in the university)
- в школе = at school
- в больнице = in/at the hospital
в университете is the prepositional case of университет:
- nominative (dictionary form): университет
- prepositional: в университете (ending -е)

в университет (accusative) would normally mean into the university (direction, movement):

Я иду в университет. – I’m going to the university.

In your sentence, we talk about where the conference will be (location, not direction), so we need в + prepositional: в университете.

Why is there no word for there like in English There will be a big scientific conference?

Russian usually doesn’t use a separate word like English there in existential sentences.

English: There will be a big scientific conference at the university soon.
Russian: В университете скоро будет большая научная конференция.

Structurally in Russian this is more like:
At the university soon will be a big scientific conference.

большая научная конференция is the grammatical subject.
будет is the verb to be in future.
в университете is where it will be.

So Russian doesn’t need an extra dummy word like there; the location phrase в университете plus будет already express existence/occurrence in a place.

Why is the verb будет used? English doesn’t say The conference will be, but There will be a conference.

Russian uses быть (будет in future) in many places where English uses there is/there will be.

есть (present) = there is/are (often omitted)
будет (future) = there will be
была/было/были (past) = there was/were

So:

В университете будет конференция.
Literally: In the university will be a conference.
Natural English: There will be a conference at the university.

Your sentence just adds скоро and adjectives:

В университете скоро будет большая научная конференция.
= There will soon be a big scientific conference at the university.

Why is большая научная конференция in the nominative case, not some other case?

Because большая научная конференция is the subject of the sentence.

The structure is:

что будет? – большая научная конференция (subject, nominative)
где? – в университете (location)
когда? – скоро (time)

In Russian, after быть (including будет) in the meaning there is/there will be, the thing that exists/occurs is normally in the nominative case:

В городе будет концерт. – There will be a concert in the city.
У нас была встреча. – We had a meeting. (literally: At us was a meeting.)

So большая научная конференция stays nominative.

Why is it большая научная конференция, not большой научной конференции?

That would be a case mismatch.

конференция is feminine, nominative singular.
Adjectives must agree with the noun in gender, number, and case.

So:

nominative feminine singular:
- большая конференция
- научная конференция
- большая научная конференция

Forms like большой научной конференции are genitive or dative or prepositional, depending on context:

нет большой научной конференции – there is no big scientific conference (genitive)
к большой научной конференции – to the big scientific conference (dative)
о большой научной конференции – about the big scientific conference (prepositional)

In your sentence the conference is the subject, so nominative большая научная конференция is correct.

Why does the adjective большая come before научная? Could I say научная большая конференция?

Both are grammatically possible, but большая научная конференция is the natural, neutral order here.

Russian tends to put:

Qualitative adjectives (size, color, personal qualities, etc.) closer to the noun.
Relative adjectives (type, domain, material, origin, like научный, literary, wooden, Russian) further from the noun.

Here:

большая – qualitative (big)
научная – relative (scientific, about science)

So the most natural order is:

большая научная конференция

научная большая конференция is possible but sounds unusual or gives special emphasis/pragmatic nuance (for example, contrasting types of conferences, focusing first on scientific, then on the fact that it’s big).

Why is it в университете скоро будет... and not скоро в университете будет... or будет скоро? How flexible is the word order?

Russian word order is flexible. All of these are possible, but the focus changes slightly:

В университете скоро будет большая научная конференция.
Neutral: first we mention the place, then the time, then what will happen.
Скоро в университете будет большая научная конференция.
Slightly more emphasis on the time – Soon, at the university there will be...
В университете будет скоро большая научная конференция.
Less common; скоро sounds somewhat more separated/emphatic:
At the university there will be, soon, a big scientific conference.
Большая научная конференция скоро будет в университете.
Often used if the conference is already known, and we’re specifying where it will be.

All are grammatically correct; the original version is a very natural, neutral-sounding order.

What exactly does скоро mean here? Is it like soon or in the near future? Are there similar words?

скоро means soon / before long / in the near future. It’s quite general and colloquial.

Nuance:

It doesn’t specify exact time – just not long from now.
Context decides whether that means minutes, days, or months.

Similar words/phrases:

вскоре – also soon, but more bookish/formal.
в ближайшее время – in the near future (more formal/neutral).
через неделю/месяц – in a week/month (more precise).

Your sentence with variants:

В университете вскоре будет большая научная конференция. – a bit more formal/literary.
В университете в ближайшее время будет большая научная конференция. – sometime soon / in the near future.

Could I say на университете instead of в университете?

Not in this meaning.

в университете – in/at the university (as an institution/place).
на университете is not idiomatic here.

With institutions, Russian often uses:

в with университет, школа, больница, библиотека:
- в университете, в школе, в больнице, в библиотеке
на with работа, почта, фабрика, завод, курсы:
- на работе, на почте, на фабрике, на заводе, на курсах

So you need в университете for at the university.

Why not say будет проходить большая научная конференция or состоится конференция? Are they different?

You can say those; they add a nuance:

будет большая научная конференция
Neutral: simply there will be a big scientific conference.
будет проходить большая научная конференция
Stresses the process / taking place over time:
a big scientific conference will be going on / will be held.
состоится большая научная конференция
More formal, emphasizes the fact of it being held / taking place as an event:
a big scientific conference will take place.

All three are correct, but:

The original sentence is the simplest neutral way to say that such a conference is planned/expected there soon.
In announcements, you’ll very often see состоится or пройдёт:
- В университете скоро состоится большая научная конференция.
- В университете скоро пройдёт большая научная конференция.

Is научная конференция exactly the same as scientific conference, or could it also mean academic conference?

научная конференция literally is scientific conference, but in practice it covers what English usually calls:

academic conference
research conference
scientific conference (in a broad sense, not only natural sciences)

It refers to a conference where researchers / scholars present and discuss their work, regardless of field:

научная конференция по истории – an academic conference on history
научная конференция по физике – a scientific conference on physics

So when translating to natural English, academic conference is often the best choice, depending on context.

How is университете pronounced and stressed?

университет (dictionary form) is stressed on the last syllable:

у-ни-вер-си-ТЕТ

In the prepositional в университете, the stress stays on the same vowel:

в у-ни-вер-си-ТЕ-те

So:

университет – [университет] (stress on е́)
в университете – [в университе́те]

The ending -е doesn’t change the stress position; it just signals the prepositional case.

Could we ever say будет большой научной конференцией instead of будет большая научная конференция?

Yes, but it would mean something different and be used in a different context.

будет большая научная конференция – there will be a big scientific conference.
Noun is in nominative as the subject.
будет большой научной конференцией – will be (serve as) a big scientific conference.
Here большой научной конференцией is instrumental case, used for predicative complement (what something is / becomes).

Example:

Это мероприятие будет большой научной конференцией.
This event will be a big scientific conference.

In your sentence, we’re not saying what something will be, but that there will be such an event, so nominative большая научная конференция is the correct form.