Poliisi ja palomies yrittävät pelastaa sekä talon ihmiset että henkilökunnan.

Because together poliisi ja palomies form a plural subject (“the police officer and the firefighter”), the verb must agree in the plural: yrittävät (3rd person plural).
Each noun (poliisi, palomies) is singular, but coordination with ja (“and”) means “two people” overall.
Using singular yrittää here would be considered grammatically wrong in standard Finnish (though in some colloquial speech you may sometimes hear mismatches).

In this sentence it clearly means a police officer (one person).
Clues:

• It’s coordinated with palomies (“firefighter”), which is one person.
• The verb yrittävät is plural because we have two individuals: one poliisi
  • one palomies.

Finnish poliisi can mean either “a police officer” (countable) or “the police” as an organization (mass noun), depending on context. Here it’s the individual person.

Pelastaa is the first infinitive (basic verb form), used here as a complement verb after yrittävät.
The structure is yrittää + infinitive = “to try to do (something)”.
So yrittävät pelastaa literally means “(they) are trying to rescue” rather than “(they) rescue”.
Only the first verb (yrittävät) is conjugated for person and number; the following verb (pelastaa) stays in the infinitive.

No, not with yrittää.
Finnish normally uses yrittää + first infinitive: yrittää pelastaa, yrittää löytää, yrittää oppia, etc.
Forms in -maan / -mään (the illative of the third infinitive, like pelastamaan) are used after other types of verbs, such as verbs of movement:

• mennä pelastamaan = “to go to rescue”
• tulla auttamaan = “to come to help”

With yrittää, stick to pelastaa.

Sekä … että … means “both … and …” and explicitly emphasizes that both listed items are included.
Here: sekä talon ihmiset että henkilökunnan = “both the people in the house and the staff”.
You could also say talon ihmiset ja henkilökunnan, and it would be grammatically fine, but sekä … että … is often a bit more formal or emphatic.
In lists of more than two, sekä often works like “as well as” or “and also”.

Talon ihmiset uses talon (genitive) as an attribute of ihmiset, literally “the house’s people”.
In Finnish, a noun in the genitive can modify another noun:

• talon katto = “the roof of the house”
• koulun oppilaat = “the pupils of the school”

So talon ihmiset = “people of the house / people in the building”.
You can’t say talossa ihmiset directly like that; to express “people who are in the house” with talossa, you’d need a relative construction, e.g. talossa olevat ihmiset.

Yes, talon is the genitive singular of talo (“house”).
In this position it shows a possessive / belonging / association relationship:

• talon ihmiset = “the house’s people” → people associated with that house.

This genitive attribute can cover many meanings: ownership, membership, location-related ties, etc. The exact nuance comes from context, but here it naturally means “the people who are in / of that building”.

They follow the general rules for total objects in Finnish:

• For plural nouns, the total object form is nominative plural:
  
  • pelastaa ihmiset (“rescue the people”) → ihmiset (nominative plural).
• For singular nouns, the total object form is genitive singular:
  
  • pelastaa henkilökunnan (“rescue the staff”) → henkilökunnan (genitive singular).

Henkilökunta (“staff”) is grammatically singular and treated like a mass/collective noun, so as a complete (total) object it appears as henkilökunnan.

Not in this sentence with this meaning.
In a normal active sentence where the whole singular object is affected, Finnish uses the genitive singular for the total object:

• Poliisi pelastaa henkilökunnan. = “The police officer rescues the staff.”

Using nominative singular henkilökunta as the object (*pelastaa henkilökunta) is not standard.
You could use henkilökuntaa (partitive) instead, but that would suggest rescuing some staff / part of the staff or an ongoing, not-necessarily-complete event.

Formally, henkilökunnan is in the genitive singular.
In traditional Finnish grammar, the accusative form of most singular nouns looks exactly like the genitive, so many books simply talk about a “genitive object” instead of separating a special accusative form.
Functionally here, henkilökunnan is the total object of pelastaa (“(they) rescue the staff completely”).
The fact that it’s an object and the verb is transitive tells you that this genitive form is behaving as a (total) object.

Finnish often avoids unnecessary repetition of a shared modifier when the meaning is clear.
Here, it’s obvious that henkilökunta is also connected to that same house/building, so talon is omitted the second time.
You could say sekä talon ihmiset että talon henkilökunnan, and it would still be correct; it just sounds a bit heavier and more explicit.
The shorter version (sekä talon ihmiset että henkilökunnan) is more natural in everyday style.

Yes, Finnish word order is flexible, as long as the parts belonging together stay together.
For example, you can front the object phrase for emphasis:

• Sekä talon ihmiset että henkilökunnan poliisi ja palomies yrittävät pelastaa.
  (“Both the people in the house and the staff the police officer and the firefighter are trying to rescue.” – with strong emphasis on both groups.)

But you cannot split sekä from its partner että, or mix the coordinated objects into the subject:

• *Poliisi ja palomies sekä talon ihmiset että henkilökunnan yrittävät pelastaa is ungrammatical/confusing.

In the original sentence, the order is neutral and clear: Subject → Verb → Object phrase.