Undersökningen börjar med tassen.

In Swedish, the definite article is usually attached to the end of the noun instead of being a separate word.

• undersökning = examination / investigation
• undersökningen = the examination
• tass = paw
• tassen = the paw

So Swedish often says in one word what English says in two.

Undersökningen comes from:

• under-söka = to examine / investigate
• undersökning = examination / investigation
• undersökningen = the examination / the investigation

The ending -en here marks the singular definite form of a common-gender noun.

Because tassen is the definite singular form: the paw.

• en tass = a paw
• tassen = the paw

If you said Undersökningen börjar med tass, that would be ungrammatical in normal Swedish.

Because tassen already means the paw, not a paw.

If you wanted a paw, you would say en tass.
So:

• med en tass = with a paw
• med tassen = with the paw

Swedish verbs do not change according to person the way English verbs do.

For example, with börja:

• jag börjar = I begin
• du börjar = you begin
• han börjar = he begins
• vi börjar = we begin

The verb form stays börjar in the present tense.

Börjar is the present tense of börja.

It can correspond to English:

• begins
• is beginning
• sometimes even does begin, depending on context

In this sentence, Undersökningen börjar med tassen, it most naturally means The examination begins with the paw.

Med means with.

In this sentence, it shows what the examination starts with:

• börjar med = begins with / starts with

This is a very common combination in Swedish:

• Filmen börjar med en sång. = The film begins with a song.
• Lektionen börjar med grammatik. = The lesson begins with grammar.

This sentence has the basic Swedish main-clause word order:

Subject + verb + other information

• Undersökningen = subject
• börjar = verb
• med tassen = prepositional phrase

So it is a very straightforward sentence.

Swedish main clauses follow the V2 rule, which means the finite verb comes in the second position. In this sentence, the subject is first, so the verb naturally comes next.

Yes. Swedish can move that part to the front, but then the verb must still stay in second position:

• Med tassen börjar undersökningen.

That is grammatically correct, but it sounds more marked or stylistic.
The original sentence, Undersökningen börjar med tassen, is the more neutral version.

Not always. Undersökning can mean several related things, depending on context:

• examination
• inspection
• investigation
• survey in some contexts

In this sentence, because of tassen (the paw), it sounds like a physical examination of an animal.

A rough guide is BUR-yar, but with Swedish sounds:

• ö is not the same as English u or o
• j in Swedish sounds like English y
• the rj combination is smoothed together in normal speech

So börjar is approximately BUR-yar, though the vowel is more rounded than in English.

Because tass is an en-word in Swedish, also called a common-gender noun.

So you get:

• en tass
• tassen

Not all nouns follow the same pattern, so this is something you often need to learn together with the noun.