Урегулирование убытка началось сегодня.

Урегулирование is a noun, not a verb. More specifically, it is a verbal noun formed from урегулировать.

In English, we often prefer a verb, like settling or resolving. Russian official style often uses nouns like this instead:

• урегулировать убыток = to settle a loss / claim
• урегулирование убытка = settlement of a loss / claim

In this sentence, урегулирование is the subject of the sentence.

It is also a neuter singular noun, which matters for the verb form началось.

Because убытка is in the genitive singular.

With a verbal noun like урегулирование, Russian often puts the related noun in the genitive:

• урегулирование чего? → убытка

Compare:

• урегулирование убытка = settlement of the loss/claim
• обсуждение вопроса = discussion of the question
• решение проблемы = solution of the problem

But with a normal finite verb, you would usually get the direct object in the accusative:

• урегулировать убыток

So this is a very common pattern:

• verb + accusative
• related noun + genitive

Because the subject is урегулирование, and that noun is neuter singular.

In the past tense, Russian verbs agree with the subject in gender and number:

• masculine: начался
• feminine: началась
• neuter: началось
• plural: начались

Since урегулирование is neuter singular, the sentence needs:

• Урегулирование ... началось

Here -ся is part of the verb начаться, which means to begin or to start.

This is different from начать:

• начать = to start something
• начаться = to start, to begin

So:

• Компания начала урегулирование убытка = The company started the settlement process.
• Урегулирование убытка началось = The settlement process began.

In your sentence, the focus is not on who started it, but on the fact that the process itself began.

Because началось is perfective, and it presents the start as a single completed event: it started today.

• началось = it started / began
• начиналось = it was starting / used to start / was beginning

In a simple announcement or factual statement, началось is the natural choice.

Начиналось would usually need a different context, for example if you were describing a process in the background or telling a story:

• Все начиналось очень спокойно. = Everything was beginning very calmly.

So in this sentence, началось is the normal form.

No, Russian word order is fairly flexible.

The sentence as given:

• Урегулирование убытка началось сегодня.

is a neutral, natural order.

But these are also possible:

• Сегодня началось урегулирование убытка.
• Урегулирование убытка сегодня началось.

The difference is mostly emphasis:

• Сегодня началось урегулирование убытка emphasizes today
• Урегулирование убытка началось сегодня sounds neutral and straightforward

Because Russian uses case endings, word order is often used more for information structure than for basic grammar.

Yes. It sounds formal, bureaucratic, or administrative.

This kind of wording is very common in:

• insurance
• legal language
• business communication
• official reports

A more everyday sentence would probably be less abstract and less noun-heavy.

Depending on the situation, a speaker might say something like:

• Сегодня начали разбираться с этим убытком.
• Сегодня начали оформление.
• Сегодня начали заниматься этим случаем.

So your sentence is correct and natural, but it belongs to a more official register.

Russian has no articles.

So Russian does not have separate words for the or a/an. Whether something is definite or indefinite is understood from context.

That means:

• Урегулирование убытка началось сегодня.

can correspond to English as:

• The settlement of the loss began today
• Settlement of the loss began today
• sometimes even A loss-settlement process began today, if the context allows it

Usually the surrounding context tells you which English article makes sense.

The stress pattern is:

Урегули́рование убы́тка начало́сь сего́дня.

A few useful notes:

• урегули́рование → stress on ли
• убы́тка → stress on бы
• начало́сь → stress on ло
• сего́дня → stress on го

One especially useful pronunciation point:

• In сегодня, the г is pronounced like в in normal speech.

So сего́дня sounds roughly like сево́дня.

Because Russian, especially in formal style, very often prefers abstract nouns where English would use a verb or a more dynamic structure.

So instead of something like:

• They started settling the claim today

Russian may package the idea as:

• Урегулирование убытка началось сегодня.

This style is common in official Russian and is worth getting used to. English usually sounds more natural with a verb, while Russian official language often sounds more natural with a noun plus a simple verb like начаться, проводиться, завершиться, and so on.