Оценщик уже здесь, оценка ущерба началась; нужно уведомить страховщика.

They come from the same root оцен-, related to evaluating or assessing, but they are different parts of speech:

• оценщик = the person who does the assessment, an assessor/appraiser
• оценка = the assessment/evaluation itself

This kind of word family is very common in Russian. A learner will often see several related words built from one root, each with a different suffix and role.

Because ущерба is in the genitive case. After a noun like оценка meaning assessment of ..., Russian usually puts the thing being assessed in the genitive:

• оценка ущерба = assessment of damage
• оценка стоимости = assessment of value
• оценка ситуации = assessment of the situation

So this is a very standard pattern: noun + genitive to express of.

Because the subject is оценка, which is a feminine singular noun. In the past tense, Russian verbs agree with the subject in gender and number:

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

Since оценка is feminine, the verb must be началась.

The verb here is начаться, not начать.

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

So началась means began / started in the sense that the process itself began. Russian often uses this reflexive-looking form when something begins on its own as an event:

• встреча началась = the meeting started
• работа началась = the work started

It is not really reflexive in the English sense; it is just part of the verb form.

This is an aspect question.

• начаться is perfective
• начинаться is imperfective

Here, началась presents the beginning as a completed fact: the assessment has now started. That is the natural choice when reporting a new state of affairs.

Начиналась would usually be used in a different context, for example for background description or repeated situations, such as оценка ущерба начиналась каждый раз с осмотра.

Because нужно can form an impersonal construction. Russian often says what is necessary without naming who must do it:

• нужно сделать = it is necessary to do
• нужно позвонить = it is necessary to call
• нужно уведомить страховщика = the insurer needs to be notified / we need to notify the insurer

English often supplies a subject like we, but Russian does not have to. The responsible person is understood from context.

It is a direct object, so it is in the accusative case. But for animate masculine singular nouns, the accusative looks the same as the genitive:

• nominative: страховщик
• accusative: страховщика

So уведомить страховщика is exactly what you expect. This is a very important rule in Russian: animate masculine nouns change form in the accusative.

Again, this is about aspect.

• уведомить = perfective, to notify once, as a completed action
• уведомлять = imperfective, to be notifying / to notify repeatedly / in general

In this sentence, the idea is that a specific notification now needs to happen, so уведомить is the natural choice. Russian often uses the perfective infinitive after нужно when talking about a single necessary action.

The sentence contains three related parts:

• Оценщик уже здесь
• оценка ущерба началась
• нужно уведомить страховщика

The first two are very closely linked, so a comma works naturally. The semicolon before the last part creates a stronger break and helps show that the final clause is more like a consequence or next required step.

Russian also often links clauses without conjunctions like and or so when the relationship is clear from context.

Yes. Russian word order is flexible, but different orders sound more or less neutral and can shift emphasis.

• Оценщик уже здесь = neutral
• Уже здесь оценщик = marked, with stronger emphasis on already here
• Оценщик здесь уже = possible, often more conversational

The original order is a very normal way to present the information.

Страховщик usually means the insurer in the formal sense — the company or legal party providing insurance. In real-life use, though, it can sometimes refer more loosely to the insurance side that needs to be informed.

If you want to be more explicit, Russian could also say:

• страховую компанию = the insurance company
• представителя страховщика = the insurer’s representative

So страховщика here is grammatically straightforward, but the exact real-world referent depends on context.