Learn Hardcore Russian: Наша компания платит большую сумму за участие в конференции. - Our company pays a large amount for participation in the conference.

In this sentence, платит is a normal transitive verb that takes a direct object (what you pay) and a prepositional phrase (what you pay for):

платит (что?) – большую сумму
платит (за что?) – за участие в конференции

Russian does not use an infinitive after за in this meaning, so за участвовать is ungrammatical. Instead, Russian prefers a verbal noun: за участие (“for participation”), not “for to participate”.

What is the difference between платит and заплатит / заплатила here?

Платит is imperfective and present tense. In this sentence it usually means:

a regular/habitual action: “Our company pays a large amount (whenever we take part).”
or a process that is seen as ongoing / not focused on result.

Заплатить is perfective and focuses on one completed payment:

Наша компания заплатит большую сумму… – “Our company will pay a large amount (once, in the future).”
Наша компания заплатила большую сумму… – “Our company paid a large amount (once, in the past).”

In English we say “is paying” and “pays”. Does платит include both meanings?

Yes. Russian has only one present tense form per aspect. Платит can mean:

“is paying” (right now, in progress), or
“pays” (in general/habitually),

depending on context. The aspect (imperfective платить vs. perfective заплатить) is more important than the English-style “simple vs. continuous” distinction.

Why is it большую сумму, not большая сумма?

Because сумму is in the accusative case as the direct object of платит.

сумма – nominative (used for subject)
сумму – accusative (used here as direct object)

The adjective большой must agree with the noun:

nominative feminine singular: большая сумма
accusative feminine singular: большую сумму

So with сумму you must say большую сумму.

Can I say just Наша компания платит много за участие в конференции instead of большую сумму?

Yes, that is correct and natural, but the nuance changes slightly.

большую сумму – sounds a bit more formal and concrete: “a large sum (of money)”.
много – more general: “a lot”.

You can also say:

Наша компания платит большие деньги за участие в конференции. – “pays a lot of money…”

All are acceptable; большую сумму is slightly more “official / business-like”.

Why is сумму in the accusative case?

Сумму is the direct object of the verb платит – it answers the question:

платит (что?) – сумму

In Russian, the direct object of a transitive verb is normally in the accusative case, so сумма (nom.) becomes сумму (acc.).

Why do we use за участие, not для участия or на участие?

These three prepositions give different meanings:

за участие (accusative) – what you pay for / in exchange for.
Here: the participation is the reason you are paying: “as payment for participation”.
для участия (genitive) – “for the purpose of participating / for participation (as a goal)”.
Example: документы для участия в конференции – “documents for participation”.
на участие (accusative) – money allocated towards participation.
Example: выделили деньги на участие в конференции – “allocated money to (be used for) participation”.

In your sentence you mean “pays a large sum for participating (as a fee)”, so за участие is the natural choice.

Why is участие in this form? Is it also accusative?

Yes. Учáстие is a neuter noun (like здание):

nominative singular: участие
accusative singular: участие (same form as nominative)

The preposition за in this meaning (“in exchange for, as payment for”) takes the accusative, so:

за (что?) – за участие.

Why is it в конференции, not в конференцию?

Because the verb is implied to be участие (в чём?) – “participation in something”, which requires the prepositional case after в:

участвовать в конференции – participate in the conference.
за участие в конференции – for participation in the conference.

В конференцию (accusative) would usually mean movement “into the conference” and is not used with участвовать / участие.

Why is it в конференции, not на конференции?

With the noun конференция, both в конференции and на конференции can appear, but they are used differently:

The verb участвовать normally governs в + prepositional:
- участвовать в конференции – participate in the conference.
На конференции is more about location:
- Я был на конференции – I was at the conference.

Since the core idea here is participation, the construction is участие в конференции, not участие на конференции.

Why is конференции spelled with -и at the end, and what case is it?

Конференции here is prepositional singular:

nominative singular: конференция
prepositional singular: о конференции, в конференции

Feminine nouns ending in -я typically have -и in the prepositional singular, so конференция → конференции after в when it means “in / about” something.

Could we say Наша компания платит большую сумму за конференцию? What would change?

You can say it, but the meaning changes:

за участие в конференции – for participation in the conference (specifically the participation fee).
за конференцию – “for the conference” in a very broad sense (e.g. paying for the event itself, organizing it, renting rooms, etc.).

If you mean a registration / participation fee, за участие в конференции is more precise and idiomatic.

Why is компания treated as singular (платит), even though a company is many people?

In Russian, компания is grammatically singular feminine, so verbs and adjectives agree with it as singular:

компания платит, компания решила, наша компания большая.

Even if the company consists of many people, it is seen as one entity grammatically, just like English “the company pays” (not “the company pay” in standard American English).