Min surprizis, ke ŝi rifuzis mian inviton.

Because min is the accusative form of mi (I / me).

In this sentence, I am the one being surprised, so I am the object of surprizis. Esperanto marks the direct object with -n:

• mi = I
• min = me

So:

• Min surprizis ... = ... surprised me

A very literal breakdown is:

• Min surprizis = [It] surprised me
• ke ŝi rifuzis mian inviton = that she refused my invitation

So the whole sentence is built like That she refused my invitation surprised me.

The subject is the whole ke-clause:

ke ŝi rifuzis mian inviton

That entire clause functions as the thing that caused the surprise.

So structurally, Esperanto is doing this:

• [Ke ŝi rifuzis mian inviton] surprizis min.
• [That she refused my invitation] surprised me.

The sentence is just written with min first:

• Min surprizis, ke ŝi rifuzis mian inviton.

That word order is perfectly normal, but the grammatical subject is still the ke-clause.

Ke means that and introduces a subordinate clause.

Here it introduces the content of what caused the surprise:

• ke ŝi rifuzis mian inviton = that she refused my invitation

This is very common in Esperanto:

• Mi scias, ke li venos. = I know that he will come.
• Estas klare, ke ŝi pravas. = It is clear that she is right.

So in your sentence, ke is just the standard way to introduce a that-clause.

Because Esperanto commonly separates subordinate clauses with commas, and a clause introduced by ke is a subordinate clause.

So:

• Min surprizis, ke ŝi rifuzis mian inviton.

This is standard punctuation in Esperanto.

You may sometimes see slight variation in punctuation style in real writing, but using a comma before a ke clause is very normal and safe for learners.

Because -is marks the past tense in Esperanto.

• surprizis = surprised
• rifuzis = refused

So both actions are being presented as past events.

Esperanto tense endings are very regular:

• -as = present
• -is = past
• -os = future
• -us = conditional
• -u = volitive/command

Here, the sentence is talking about a past refusal and a past reaction to it, so -is appears on both verbs.

Because inviton is the direct object of rifuzis, and adjectives in Esperanto agree with the nouns they describe.

So:

• invito = invitation
• inviton = invitation, as a direct object
• mia = my
• mian = my, agreeing with an accusative noun

Since rifuzis takes a direct object, the invitation gets -n, and the adjective mia also gets -n to match:

• mian inviton = my invitation

This agreement is a very important Esperanto rule:

• bela domo = a beautiful house
• belan domon = a beautiful house (as object)

Because possessives like mia, via, ŝia, etc. already make the noun definite enough.

So Esperanto normally says:

• mia libro = my book
• ŝia domo = her house
• mian inviton = my invitation

You generally do not say la mia invito in this ordinary sense.

That construction can exist in special contexts, but it does not mean the same simple thing as English my invitation. For a normal possessive, just use mia/mian without la.

Yes. That version is grammatically correct too.

Both mean the same thing:

• Min surprizis, ke ŝi rifuzis mian inviton.
• Ke ŝi rifuzis mian inviton surprizis min.

The difference is mainly word order and emphasis.

The original sentence starts with min, which puts early focus on me and sounds natural in Esperanto. The alternative starts with the whole that-clause, which can feel a bit heavier because it begins with a long subject.

So the original wording is often smoother, but both are correct.

Yes, and that is a very natural alternative.

There is a useful difference:

• Min surprizis, ke ŝi rifuzis mian inviton.
  = It surprised me that she refused my invitation.

• Mi surpriziĝis, ke ŝi rifuzis mian inviton.
  = I became surprised / I was surprised that she refused my invitation.

The first uses the transitive verb surprizi (to surprise): something surprised me.

The second uses surpriziĝi, with -iĝ-, which often means to become / to get into a state: I became surprised.

In many contexts, these are very close in meaning, and both are idiomatic.

Ŝi is pronounced roughly like shee in English.

• ŝ sounds like English sh
• i sounds like ee

So:

• ŝi ≈ shee

More broadly, a few useful pronunciation points from the sentence are:

• ŝi = shee
• rifuzis = ree-foo-ZEE-sis
• mian = roughly MEE-an
• inviton = in-vee-TOHN

And remember that Esperanto stress normally falls on the second-to-last syllable:

• sur-PRI-zis
• ri-fu-ZIS
• in-vi-TON

Usually, yes: surprizi is a transitive verb meaning to surprise someone.

So:

• Tio surprizis min. = That surprised me.
• La novaĵo surprizis ŝin. = The news surprised her.

In your sentence, the ke-clause plays the role that tio or la novaĵo could play:

• Ke ŝi rifuzis mian inviton surprizis min.

So if you remember surprizi iun = to surprise someone, the structure becomes easier to understand.