Tapasin hänen vaimonsa eilen.

The dictionary form is tavata, meaning to meet.

Tapasin is the 1st person singular past tense form, so it means I met.
This verb looks a little tricky because its stem changes: tavata → tapasin. That is normal for this verb.

Because Finnish usually does not need a subject pronoun when the verb ending already shows who the subject is.

Here, the ending -n in tapasin tells you the subject is I. So minä is unnecessary unless you want emphasis:

• Tapasin hänen vaimonsa eilen. = neutral
• Minä tapasin hänen vaimonsa eilen. = more emphatic, like I met his/her wife yesterday

Hänen is the genitive form of hän.

• hän = he / she
• hänen = literally the possessive form, used like his / her

In Finnish, possession is often expressed with a genitive possessor plus a possessive suffix on the noun, which is exactly what you see in hänen vaimonsa.

It can mean either.

Finnish does not normally distinguish gender in third-person singular pronouns:

• hän = he or she
• hänen = his or her

So only the context tells you whether the wife belongs to a man or a woman.

Yes, they both contribute to showing possession.

In standard Finnish, third-person possession is commonly marked with:

• the possessor in the genitive: hänen
• a possessive suffix on the noun: vaimonsa

So:

• hänen vaimonsa = his/her wife

This is normal standard Finnish.
In everyday speech, people often simplify things, but the full written standard form uses both.

Also, the suffix has two forms depending on vowel harmony:

• -nsa
• -nsä

Here it is -nsa because vaimo has back vowels.

Because the noun has the 3rd person possessive suffix attached:

• vaimo = wife
• vaimonsa = his/her wife

A useful thing to know is that with possessive suffixes, the visible form of a singular noun often does not look different for nominative, genitive, and accusative the way you might expect.

So even though this noun is the object of the sentence, vaimonsa is still the correct form on the surface.

That is why you do not see a separate -n as in plain vaimon.

Because this sentence presents the meeting as a complete, finished event.

In Finnish, a completed affirmative action like this usually takes a total object, not a partitive object. So hänen vaimonsa is the normal choice here.

A partitive object would suggest something less complete or a different kind of meaning, and it would not be the normal form with tavata in a simple sentence like this.

Eilen is a time adverb, meaning yesterday.

Yes, it can move. Finnish word order is fairly flexible. The sentence as given is a natural neutral order:

• Tapasin hänen vaimonsa eilen.

But you could also say:

• Eilen tapasin hänen vaimonsa.

That version puts more focus on yesterday.

So the meaning stays basically the same, but the information focus can shift.

Sometimes yes, but context matters.

Vaimonsa alone can mean his/her wife, if the possessor is already clear from the context.

However, hänen vaimonsa is clearer, and in a sentence like this it is the safest standard way to show that the wife belongs to someone else.

This matters because Finnish often uses possessive suffixes in a reflexive-like way with the subject:

• Hän tapasi vaimonsa.
  This usually means He/She met his/her own wife.

So in your sentence, hänen vaimonsa makes the possessor explicit and avoids confusion.

Not as much as in English.

Finnish relies more on case endings and verb forms than on fixed word order, so the structure is more flexible. The given order is natural and neutral, but other orders are possible for emphasis.

For example:

• Tapasin hänen vaimonsa eilen. = neutral
• Eilen tapasin hänen vaimonsa. = emphasizes yesterday
• Hänen vaimonsa tapasin eilen. = emphasizes his/her wife

So the basic meaning stays the same, but the focus changes.