Mit der Wasserwaage prüfen wir, ob das Brett waagerecht ist.

Because mit always takes the dative case in German.

• Base form: die Wasserwaage
• After mit: der Wasserwaage

So mit der Wasserwaage means with the spirit level / with the level.

A useful rule:

• mit + dative
• für + accusative
• ohne + accusative
• zu + dative

So this is not a special feature of this sentence; it is a general rule triggered by mit.

Because German main clauses normally follow the verb-second rule.

The first position in the sentence is occupied by Mit der Wasserwaage. Once that happens, the finite verb must come next:

• Mit der Wasserwaage | prüfen | wir, ...

So the subject wir comes after the verb.

Compare:

• Wir prüfen mit der Wasserwaage, ob das Brett waagerecht ist.
• Mit der Wasserwaage prüfen wir, ob das Brett waagerecht ist.

Both are correct. The version with Mit der Wasserwaage first puts more focus on the tool being used.

Ob means whether / if when introducing an indirect yes-no question.

In this sentence, the thing being checked is a yes-no question:

• Is the board level?
• Yes or no?

That is why German uses ob:

• ..., ob das Brett waagerecht ist.

Compare:

• Ich weiß, ob er kommt. = I know whether he is coming.
• Ich weiß, dass er kommt. = I know that he is coming.

So:

• ob = whether / if
• dass = that

You use ob when the embedded idea could be answered with yes or no.

Because ob introduces a subordinate clause, and in German subordinate clauses, the conjugated verb usually goes to the end.

Main clause:

• Wir prüfen ...

Subordinate clause:

• ..., ob das Brett waagerecht ist.

That final ist is completely normal German word order.

Compare:

• Das Brett ist waagerecht.
  This is a main clause.

• ..., ob das Brett waagerecht ist.
  This is a subordinate clause, so ist moves to the end.

Because waagerecht is being used as a predicate adjective, after the verb sein.

In German, adjectives used this way do not take adjective endings:

• Das Brett ist waagerecht.
• Die Wand ist gerade.
• Der Tisch ist groß.

But when an adjective comes before a noun, it does take an ending:

• ein waagerechtes Brett
• eine große Wand
• ein gerader Schnitt

So:

• after sein: waagerecht
• before a noun: waagerechtes Brett

Das Brett is in the nominative case because it is the subject of the clause ob das Brett waagerecht ist.

The clause is basically:

• Das Brett ist waagerecht.

Here, das Brett is the thing that is level, so it is the subject.

Also, Brett is a neuter noun, which is why the nominative singular article is das.

Wasserwaage is a compound noun:

• Wasser = water
• Waage = scale / balance

In German, the last part of a compound usually determines the gender. Since die Waage is feminine, die Wasserwaage is also feminine.

That is why you get:

• nominative: die Wasserwaage
• dative after mit: der Wasserwaage

This is a very useful rule for compound nouns in German.

Yes. That version is also correct.

German allows some flexibility in word order, especially with adverbials and prepositional phrases. The difference is mainly one of focus or emphasis:

• Wir prüfen mit der Wasserwaage, ob das Brett waagerecht ist.
  More neutral.

• Mit der Wasserwaage prüfen wir, ob das Brett waagerecht ist.
  More emphasis on how we are checking it.

So the sentence you were given is not strange; it just foregrounds the instrument.

They are very close in meaning, but waagerecht is often the more natural everyday word when talking about whether something is level.

• waagerecht = level, horizontal
• horizontal = horizontal, but often a bit more technical or formal

With a board and a spirit level, waagerecht sounds very natural.

You may also see:

• senkrecht = vertical
• horizontal = horizontal
• waagerecht = level / horizontal

In practical contexts like building, measuring, or hanging shelves, waagerecht is very common.

It is a prepositional phrase that tells you the means/instrument used to perform the action.

So it answers the question:

• How do we check?
• With the spirit level.

This is similar to English expressions like:

• with a ruler
• with a hammer
• with a flashlight

In German:

• mit einem Lineal
• mit einem Hammer
• mit einer Taschenlampe
• mit der Wasserwaage

So here it functions like an instrumental phrase, even though German does not have a separate instrumental case. It uses mit + dative instead.